Giving More Realistic Definitions of Trigonometric Ratios

ERIC Educational Resources Information Center

Bhattacharjee, Pramode Ranjan

2012-01-01

Trigonometry is a well known branch of Mathematics. The study of trigonometry is of great importance in surveying, astronomy, navigation, engineering, and in different branches of science. This paper reports on the discovery of flaws in the traditional definitions of trigonometric ratios of an angle, which (in most cases) make use of the most…

Mathematics Teacher-Candidates' Performance in Solving Problems with Different Representation Styles: The Trigonometry Example

ERIC Educational Resources Information Center

Dündar, Sefa

2015-01-01

Using multiple representations of a problem can reveal the relationship between complex concepts by expressing the same mathematical condition differently and can contribute to the meaningful learning of mathematical concepts. The purpose of this study is to assess the performances of mathematics teacher-candidates on trigonometry problems…

Students' Perceptions and Development of Conceptual Understanding Regarding Trigonometry and Trigonometric Function

ERIC Educational Resources Information Center

Cetin, Omer Faruk

2015-01-01

This study aims to analyse university level mathematics education students' perceptions on conceptual understanding of trigonometry and trigonometric functions and their content development of these concepts. A case study was conducted with 90 freshman students of Elementary Mathematics Department. The data were gathered via a scale; they included…

Trigonometry with Year 8: Part 1

ERIC Educational Resources Information Center

Steer, Jessica; de Vila, Maria Antioneta; Eaton, James

2009-01-01

The authors explore the teaching of trigonometry using a method developed by Jeremy Burke of Kings College. A series of lessons was planned using an approach which looks at moving from a mathematical description of the topic, to a sequence plan, to a set of activities, which students can use to help them come to understand the topic. This is…

A Qualitative Study Comparing the Instruction on Vectors between a Physics Course and a Trigonometry Course

ERIC Educational Resources Information Center

James, Wendy Michelle

2013-01-01

Science and engineering instructors often observe that students have difficulty using or applying prerequisite mathematics knowledge in their courses. This qualitative project uses a case-study method to investigate the instruction in a trigonometry course and a physics course based on a different methodology and set of assumptions about student…

Enhancing Conceptual Understanding of Trigonometry Using Earth Geometry and the Great Circle

ERIC Educational Resources Information Center

Wongapiwatkul, Pimpalak; Laosinchai, Parames; Panijpan, Bhinyo

2011-01-01

Trigonometry is an integral part of the draft for the Senior Secondary Australian National Curriculum for Mathematics, as it is a topic in Unit 2 of both Specialist Mathematics and Mathematics Methods, and a reviewing topic in Unit 1, Topic 3: Measurement and Geometry of General Mathematics. However, learning trigonometric ideas is difficult for…

Interactive Computer-Supported Learning in Mathematics: A Comparison of Three Learning Programs on Trigonometry

ERIC Educational Resources Information Center

Sander, Elisabeth; Heiß, Andrea

2014-01-01

Three different versions of a learning program on trigonometry were compared, a program controlled, non-interactive version (CG), an interactive, conflict inducing version (EG 1), and an interactive one which was supposed to reduce the occurrence of a cognitive conflict regarding the central problem solution (EG 2). Pupils (N = 101) of a…

Comparative Effects of Concept Mapping and Cooperative Learning Strategies on Senior Secondary School Students' Achievement in Mathematics-Trigonometry in Kano State, Nigeria

ERIC Educational Resources Information Center

Bot, Thomas D.; Eze, John E.

2016-01-01

This article presents the findings from an experimental study on the effectiveness of concept mapping and cooperative learning strategies on SSII students' achievement in trigonometry in mathematics. The research design used in conducting the study was quasi-experimental pre-test and post-test non-equivalent control group. The sample consisted of…

What Subject Matter Knowledge Do Second-Level Teachers Need to Know to Teach Trigonometry? An Exploration and Case Study

ERIC Educational Resources Information Center

Walsh, Richard; Fitzmaurice, Olivia; O'Donoghue, John

2017-01-01

This study reports on the level of trigonometry Subject Matter Knowledge (SMK) of third and final-year pre-service second-level mathematics teachers () at an Irish third-level institution. The aim of the study was to determine if this sample of prospective teachers has an appropriate level of SMK to teach second-level trigonometric concepts. The…

Analysis of the Cognitive Unity or Rupture between Conjecture and Proof When Learning to Prove on a Grade 10 Trigonometry Course

ERIC Educational Resources Information Center

Fiallo, Jorge; Gutiérrez, Angel

2017-01-01

We present results from a classroom-based intervention designed to help a class of grade 10 students (14-15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students' solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs.…

A qualitative study comparing the instruction on vectors between a physics course and a trigonometry course

NASA Astrophysics Data System (ADS)

James, Wendy Michelle

Science and engineering instructors often observe that students have difficulty using or applying prerequisite mathematics knowledge in their courses. This qualitative project uses a case-study method to investigate the instruction in a trigonometry course and a physics course based on a different methodology and set of assumptions about student learning and the nature of mathematics than traditionally used when investigating students' difficulty using or applying prerequisite mathematics knowledge. Transfer theory examined within a positivist or post-positivist paradigm is often used to investigate students' issue applying their knowledge; in contrast, this qualitative case-study is positioned using constructionism as an epistemology to understand and describe mathematical practices concerning vectors in a trigonometry and a physics course. Instructor interviews, observations of course lectures, and textbooks served as the qualitative data for in-depth study and comparison, and Saussure's (1959) concept of signifier and signified provided a lens for examining the data during analysis. Multiple recursions of within-case comparisons and across-case comparison were analyzed for differences in what the instructors and textbooks explicitly stated and later performed as their practices. While the trigonometry and physics instruction differed slightly, the two main differences occurred in the nature and use of vectors in the physics course. First, the "what" that is signified in notation and diagrams differs between contextualized and context-free situations, and second, physics instruction taught vectors very similar to trigonometry instruction when teaching the mathematics for doing physics, but once instruction focused on physics, the manner in which vector notation and diagrams are used differed from what is explicitly stated during mathematics instruction.

A Vector Approach to Euclidean Geometry: Inner Product Spaces, Euclidean Geometry and Trigonometry, Volume 2. Teacher's Edition.

ERIC Educational Resources Information Center

Vaughan, Herbert E.; Szabo, Steven

This is the teacher's edition of a text for the second year of a two-year high school geometry course. The course bases plane and solid geometry and trigonometry on the fact that the translations of a Euclidean space constitute a vector space which has an inner product. Congruence is a geometric topic reserved for Volume 2. Volume 2 opens with an…

Using Open-Response Tasks to Reveal the Conceptual Understanding of Learners--Learners Teaching the Teacher What They Know about Trigonometry

ERIC Educational Resources Information Center

Price, Charmaine; van Jaarsveld, Pieter

2017-01-01

This article reports on using open-response questions to reveal the level of, and change in, conceptual understanding of a small sample of Grade 11 learners of trigonometry in a South African high school. The investigation used learner response sheets in a regular classroom with the teacher as researcher. Combining the idea of concept image and…

Analyzing and Comparing the Two Grade-Ten Mathematics Textbooks Versions (Arabic and English) Used in Abu Dhabi Schools

ERIC Educational Resources Information Center

Abosalem, Yousef

2016-01-01

This study was conducted to compare two grade-ten mathematics textbooks according to Bloom's Taxonomies. In the Arabic version, 37 out of 70 periods (55.29%) were given to plane geometry and trigonometry, whereas 29 out of 70 periods (41.41%) were allocated for geometry and trigonometry. Also, 12 periods (17.14%) were allocated for algebra in the…

The Trigonometry of Twistors and Elementary Particles

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gustafson, Karl

2009-03-10

A new trigonometry for twistors is presented. The operator-theoretic maximum twistor turning angle is shown to be related to the space-time geometric angle within the light cone. The corresponding maximally turned twistor antieigenvectors are calculated and interpretted. The two weak interaction CP eigenvectors of neutral kaons are shown to be exactly the two strong interaction strangeness antieigenvectors. Quark mixing is seen trigonometrically. 't Hooft's microcosmos model is connected to the theories of normal degree and complex dynamics.

Summer Research Apprentice Program report. [Summer Research Apprentice Program

DOE Office of Scientific and Technical Information (OSTI.GOV)

Curington, B.

1982-01-01

The Summer Research Apprentice Program is designed to provide students with their first look at college life while preparing them for possible careers in mathematics, science and engineering. The 23 students, enrolled as college freshmen for 8 hours of college credit, took courses in Trigonometry, College Algebra and introduction to Research (4 students were enrolled in Calculus 1 instead of Trigonometry and College Albebra). During this third year of operation, refinements were made in both the administration of the program and in the method of implementation.

The D-dimensional non-relativistic particle in the Scarf Trigonometry plus Non-Central Rosen-Morse Potentials

NASA Astrophysics Data System (ADS)

Deta, U. A.; Lestari, N. A.; Yantidewi, M.; Suparmi, A.; Cari, C.

2018-03-01

The D-Dimensional Non-Relativistic Particle Properties in the Scarf Trigonometry plus Non-Central Rosen-Morse Potentials was investigated using an analytical method. The bound state energy is given approximately in the closed form. The approximate wave function for arbitrary l-state in D-dimensions are expressed in the form of generalised Jacobi Polynomials. The energy spectra of the particle are increased when the dimensions are higher. The relationship between the orbital number in each dimension is recursive. The special case in 3 dimensions is given to the ground state.

Laws of trigonometry on SU(3)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Aslaksen, H.

1988-01-01

In this paper we will study triangles in SU(3). The orbit space of congruence classes of triangles in SU(3) has dimension 8. Each corner is made up of a pair of tangent vectors (X,Y), and we consider the 8 functions trX{sup 2}, i trX{sup 3}, trY{sup 2}, i trY{sup 3}, trXY, i trY{sup 2}Y, i trXY{sup 2}, trX{sup 2}Y{sup 2} which are invariant under the full isometry group of SU(3). We show that these 8 corner invariants determine the isometry class of the triangle. We give relations (laws of trigonometry) between the invariants at the different corners, enabling us tomore » determine the invariants at the remaining corners, including the values of the remaining side and angles, if we know one set of corner invariants. The invariants that only depend on one tangent vector we will call side invariants, while those that depend on two tangent vectors will be called angular invariants. For each triangle we then have 6 side invariants and 12 angular invariants. Hence we need 18 {minus} 8 = 10 laws of trigonometry. The basic tool for deriving these laws is a formula expressing tr(exp X exp Y) in terms of the corner invariants.« less

Trigonometry-Integrated 'Lift' Technique (TILT) for Restoring Volar Tilt in Distal Radius Fractures: Description of Technique and Preliminary Results.

PubMed

Sechachalam, Sreedharan; Satku, Mala; Wong, Jian Hao Kevin; Tan, Lester Teong Jin; Yong, Fok Chuan

2017-03-01

Restoration of extra-articular and intra-articular parameters are important considerations during operative fixation of distal radius fractures. Restoration of volar tilt by using visual estimation and the 'lift' technique has previously been described. The aim of our study was to describe a mathematical technique for accurately restoring the volar tilt of the distal radius to acceptable anatomic values. A retrospective review of cases performed using the trigonometry-integrated ' lift' technique (TILT) was performed. This technique uses the pre-operative volar tilt angle as well as the dimensions of the implant to calculate the 'lift' required to restore volar tilt. Intra-operative angles were measured using a marked transparency overlay on fluoroscopic images. Pre-operative and post-operative volar tilt were measured and analysed. Twenty-seven fractures were included in the study, with 20 being classified as Arbeitsgemeinschaft für Osteosynthesefragen (AO) C-type. Pre-'lift' volar tilt ranged from 0° to -20°. Post-'lift' volar tilt ranged from 2° to 16°, with all but three cases ranging from 5° to 15°. The mean volar tilt achieved was 10.2°. The trigonometry-integrated 'lift' technique resulted in reliable intra-operative restoration of anatomic volar tilt in distal radius fractures.

Exploring Data: Euclid's Way.

ERIC Educational Resources Information Center

Brinkworth, Peter

1998-01-01

Introduces handling data as conceived by Euclid, which provides some interesting possibilities for students to investigate fundamental geometrical ideas as well as relating some elementary geometry with elementary trigonometry. (ASK)

Software Reviews.

ERIC Educational Resources Information Center

Mathematics and Computer Education, 1988

1988-01-01

Presents reviews of six software packages. Includes (1) "Plain Vanilla Statistics"; (2) "MathCAD 2.0"; (3) "GrFx"; (4) "Trigonometry"; (5) "Algebra II"; (6) "Algebra Drill and Practice I, II, and III." (PK)

Designs of goal-free problems for trigonometry learning

NASA Astrophysics Data System (ADS)

Retnowati, E.; Maulidya, S. R.

2018-03-01

This paper describes the designs of goal-free problems particularly for trigonometry, which may be considered a difficult topic for high school students.Goal-free problem is an instructional design developed based on a Cognitive load theory (CLT). Within the design, instead of asking students to solve a specific goal of a mathematics problem, the instruction is to solve as many Pythagoras as possible. It was assumed that for novice students, goal-free problems encourage students to pay attention more to the given information and the mathematical principles that can be applied to reveal the unknown variables. Hence, students develop more structured knowledge while solving the goal-free problems. The resulted design may be used in regular mathematics classroom with some adjustment on the difficulty level and the allocated lesson time.

Analytic Methods in Investigative Geometry.

ERIC Educational Resources Information Center

Dobbs, David E.

2001-01-01

Suggests an alternative proof by analytic methods, which is more accessible than rigorous proof based on Euclid's Elements, in which students need only apply standard methods of trigonometry to the data without introducing new points or lines. (KHR)

Can a Crescent Mars Ever Be Seen from Earth?

ERIC Educational Resources Information Center

Lamb, John F., Jr.

1990-01-01

Described is an activity that incorporates a computer, geometry, algebra, trigonometry, and calculus to answer questions about the planet Mars. A possible crescent of Mars is compared to those of Venus and Mercury. (KR)

Two Comments on Bond Angles

NASA Astrophysics Data System (ADS)

Glaister, P.

1997-09-01

Tetrahedral Bond Angle from Elementary Trigonometry The alternative approach of using the scalar (or dot) product of vectors enables the determination of the bond angle in a tetrahedral molecule in a simple way. There is, of course, an even more straightforward derivation suitable for students who are unfamiliar with vectors, or products thereof, but who do know some elementary trigonometry. The starting point is the figure showing triangle OAB. The point O is the center of a cube, and A and B are at opposite corners of a face of that cube in which fits a regular tetrahedron. The required bond angle alpha = AÔB; and using Pythagoras' theorem, AB = 2(square root 2) is the diagonal of a face of the cube. Hence from right-angled triangle OEB, tan(alpha/2) = (square root 2) and therefore alpha = 2tan-1(square root 2) is approx. 109° 28' (see Fig. 1).

[Thomas Fincke and trigonometry].

PubMed

Schönbeck, Jürgen

2004-01-01

Thomas Fincke (January 6th, 1561 - April 24th, 1650), born in Flensburg (Germany), was one of the very most important and significant scientists in Denmark during the seventeenth century, a mathematician and astrologer and physician in the beginning of modern science, a representative of humanism and an influentual academic organizer. He studied in Strasbourg (since 1577) and Padua (since 1583) and received his M.D. in Basel (1587), he practised as a physician throughtout his life (since 1587 or 1590) and became a professor at Copenhagen (1591). But he was best known because of his Geometriae rotundi libri XIIII (1583), a famous book on plane and spherical trigonometry, based not on Euclid but on Petrus Ramus. In this influentual work, in which Fincke introduced the terms tangent and secant and probable first noticed the Law of Tangents and the so-called Newton-Oppel-Mauduit-Simpson-Mollweide-Gauss-formula, he showed himself to be ,,abreast of the mathematics of his time".

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

Investigation of the Great Pyramid of Giza.

ERIC Educational Resources Information Center

Peace, Nigel; And Others

1997-01-01

Describes an activity in which geometry and trigonometry are studied using pyramids. Identical model pyramids are constructed from card stock, along with pyramids of different proportions and cuboids to use as controls. Also includes an investigation of some apparently non-scientific claims. (DDR)

Redefining a Model

ERIC Educational Resources Information Center

Hodges, Thomas E.

2007-01-01

This article describes an alternate way to utilize a circular model to represent thirds by incorporating areas of circular segments, trigonometric functions, and geometric transformations. This method is appropriate for students studying geometry and trigonometry at the high shool level. This task provides valuable learning experiences that…

Gary O's Fence Question.

ERIC Educational Resources Information Center

Daniels, David S.

1993-01-01

Discusses the problem of finding the amount of fence it would require for the outfield fence of a baseball field of given dimensions. Presents different solution methods for each of the levels from grades 9-12. The different methods incorporate geometry, trigonometry, analytic geometry, and calculus. (MDH)

Development of automatic pre-tracking system for fillet weld based on laser trigonometry

NASA Astrophysics Data System (ADS)

Shen, Xiaoqin; Yu, Fusheng

2005-01-01

In this paper, an automatic fillet weld pre-tracking system for welding the work piece of lorry back boards with several bend in haul automobile is developed basing on laser trigonometry. The optical measuring head based on laser-PSD trigonometry is used as position sensor. It is placed in front of the traveling direction of welding wire to get the distances from welding wire to the two side boards of the welding lines, upper board and bottom board of the fillet weld respectively. A chip of AT89S52 is used as the micro controller in this system. The AC servomotors, ball-screws and straight guide rails constitute the sliding table to take welding wire move. The laser-PSD sensors pass through the vertical board, upper board and bottom board of the fillet weld when welding wire moves and then get the distance. The laser-PSD sensors output the analog signals. After A/D conversion, the digital signal is input into AT89S52 and calculated. Then the information of the position and lateral deviation of the welding wire when welding a certain position are gotten to control welding wires. So the weld pre-tracking for welding the work piece with long distance and large bend in haul automobile is realized. The position information is input into EEPROM to be saved for short time after handled by AT89S52. The information is as the welding position information as well as the speed adjusting data of the welding wire when it welds the several bend of the work piece. The practice indicates that this system has high pre-tracking precision, good anti-disturb ability, excellent reliability, easy operating ability and good adaptability to the field of production.

Maximizing the Range of a Projectile.

ERIC Educational Resources Information Center

Brown, Ronald A.

1992-01-01

Discusses solutions to the problem of maximizing the range of a projectile. Presents three references that solve the problem with and without the use of calculus. Offers a fourth solution suitable for introductory physics courses that relies more on trigonometry and the geometry of the problem. (MDH)

Graphical Method for Determining Projectile Trajectory

ERIC Educational Resources Information Center

Moore, J. C.; Baker, J. C.; Franzel, L.; McMahon, D.; Songer, D.

2010-01-01

We present a nontrigonometric graphical method for predicting the trajectory of a projectile when the angle and initial velocity are known. Students enrolled in a general education conceptual physics course typically have weak backgrounds in trigonometry, making inaccessible the standard analytical calculation of projectile range. Furthermore,…

Introducing Group Theory through Music

ERIC Educational Resources Information Center

Johnson, Craig M.

2009-01-01

The central ideas of postcalculus mathematics courses offered in college are difficult to introduce in middle and secondary schools, especially through the engineering and sciences examples traditionally used in algebra, geometry, and trigonometry textbooks. However, certain concepts in music theory can be used to expose students to interesting…

Curriculum Change in Secondary School Mathematics

ERIC Educational Resources Information Center

Alspaugh, John W.; and others

1970-01-01

Discusses six major trends in mathematics curriculum development: lowering of grade placement, teaching methods from memorization to discovery, introduction and deletion of content, integration of plane and solid geometry, algebra, and trigonometry, emphasis upon needs and characteristics of student, and increasing rate of curriculum change.…

Sequencing Computer-Assisted Learning of Transformations of Trigonometric Functions

ERIC Educational Resources Information Center

Ross, John A.; Bruce, Catherine D.; Sibbald, Timothy M.

2011-01-01

Studies incorporating technology into the teaching of trigonometry, although sparse, have demonstrated positive effects on student achievement. The optimal sequence for integrating technology with teacher-led mathematics instruction has not been determined. Our research investigated whether technology has a greater impact on student achievement…

Simple trigonometry on computed tomography helps in planning renal access.

PubMed

Bilen, Cenk Yücel; Koçak, Burak; Kitirci, Gürcan; Danaci, Murat; Sarikaya, Saban

2007-08-01

To retrospectively assess the usefulness of the measurements on preoperative computed tomography (CT) of patients with urinary stone disease for planning the access site using vertical angulation of the C-arm. Of the patients who underwent percutaneous nephrolithotomy from November 2001 to October 2006, 41 patients with superior calix access had undergone preoperative CT. The depth of the target stone (y) and the vertical distance from that point to the first rib free slice (x) were measured on CT. The limit of the ratio of x over y was accepted as 0.58, with ratios below that indicating that infracostal access could be achieved by vertical angulation of the C-arm. We achieved an approach to the superior calix through an infracostal access in 28 patients. The preoperative trigonometric study on CT predicted 24 of them. The stone-free rate was 92.6%, and no chest-related complications developed. Simple trigonometry on CT of the patients with complex stones could help endourologists in planning renal access.

A Glossary for Pre-Calculus

ERIC Educational Resources Information Center

Arnold, Bruce; Kracht, Brenda; Ross, Judy; Teegarden, Terrie; Tompkins, Maurice

2012-01-01

In the deconstruction of the California state standards for trigonometry, linear algebra and mathematical analysis for the Cal-PASS (California Partnership for Achieving Student Success) Content Standards Deconstruction projects, it became apparent that terms were used for which no definition was given. The San Diego Central Cal-PASS Math…

Mapping Stars with TI-83.

ERIC Educational Resources Information Center

Felsager, Bjorn

2001-01-01

Describes a mathematics and science project designed to help students gain some familiarity with constellations and trigonometry by using the TI-83 calculator as a tool. Specific constellations such as the Big Dipper (Plough) and other sets of stars are located using stereographic projection and graphed using scatterplots. (MM)

The Monkey and the Hunter.

ERIC Educational Resources Information Center

Rogowski, Steve

1982-01-01

A problem is detailed which has a solution that embodies geometry, trigonometry, ballistics, projectile mechanics, vector analysis, and elementary computer graphics. It is felt that the information and sample computer programs can be a useful starting point for a user written code that involves missiles and other projectiles. (MP)

Imagine Yourself in This Calculus Classroom

ERIC Educational Resources Information Center

Bryan, Luajean

2007-01-01

The efforts to attract students to precalculus, trigonometry, and calculus classes became more successful at the author's school when projects-based classes were offered. Data collection from an untethered hot air balloon flight for calculus students was planned to maximize enrollment. The data were analyzed numerically, graphically, and…

Problems with Generalising: Pythagoras in N Dimensions

ERIC Educational Resources Information Center

Foster, Colin

2013-01-01

Pythagoras' theorem in two and three dimensions appears in General Mathematics, Units 1-2, section 6 (Geometry and trigonometry: Shape and measurement) in the Victorian Certificate of Education Mathematics Study Design (Victorian Curriculum Assessment Authority, 2010). It also comes in Further Mathematics, Units 3-4 (Applications: Geometry and…

46 CFR 310.55 - Scholastic requirements.

Code of Federal Regulations, 2010 CFR

2010-10-01

... SAT. A candidate electing to use the ACT, shall take all the tests, namely, English, Mathematics... Mathematics (from algebra, geometry and trigonometry); (B) 3 units of English; and (C) 1 unit of Physics or... science; (B) Foreign language; (C) Economics; and, (D) Social science. (2) Evidence of academic work...

Calculus of Elementary Functions, Part I. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics including algebra, axiomatic geometry, trigonometry, and analytic geometry. It does not assume they have acquired a background of elementary functions. This teacher's guide contains background information, suggested instructional procedures, and…

Math 3011--College Algebra and Trigonometry. Course Outline.

ERIC Educational Resources Information Center

New York Inst. of Tech., Old Westbury.

This document contains the course syllabus and 12 independent practice modules for a college level mathematics course designed to provide the necessary foundation for success in calculus, develop logical thinking skills, and enhance analytic skills through problem solving. Topics include relations and functions; inequalities; complex numbers;…

Mathematics, Vol. 2.

ERIC Educational Resources Information Center

Bureau of Naval Personnel, Washington, DC.

The second of three volumes of a mathematics training course for Navy personnel, this document contains material primarily found at the college level. Beginning with logarithms and trigonometry, the text moves into vectors and static equilibrium (physics). Coordinate geometry, conic sections, and the tangents, normals, and slopes of curves follow.…

[Charnley-type total hip prosthesis. Radiological technic of angular measurements of the acetabular piece (anteversion)].

PubMed

Chevrot, A; Najman, G

1983-01-01

A radiological technique is described based on the study of antero-posterior and lateral views of the hip. Mathematical calculations by trigonometry make it possible to deduce the degree of anteversion of the acetabular cup. The necessary tables are given.

The Spreadsheet in an Educational Setting. Microcomputing Working Paper Series F 84-4.

ERIC Educational Resources Information Center

Wozny, Lucy

This overview of a specific spreadsheet, Microsoft's Multiplan for the Apple Macintosh microcomputer, emphasizes specific features that are important to the academic community, including the mathematical functions of algebra, trigonometry, and statistical analysis. Additional features are summarized, including data formats for both numerical and…

Seasonal Variation in Epidemiology

ERIC Educational Resources Information Center

Marrero, Osvaldo

2013-01-01

Seasonality analyses are important in medical research. If the incidence of a disease shows a seasonal pattern, then an environmental factor must be considered in its etiology. We discuss a method for the simultaneous analysis of seasonal variation in multiple groups. The nuts and bolts are explained using simple trigonometry, an elementary…

Curriculum Guide for Baccalaureate Oriented Courses in Mathematics.

ERIC Educational Resources Information Center

Darnes, G. Robert, Ed.

A mathematics curriculum guide is presented for the purpose of offering statewide guidelines to colleges for determining the content of those courses which might be considered standard courses in the first two years of the college curriculum. Courses covered include: intermediate algebra, college algebra, trigonometry, analytic geometry,…

Core-Plus Mathematics. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2010

2010-01-01

"Core-Plus Mathematics" is a four-year curriculum that replaces the traditional sequence with courses that each feature interwoven strands of algebra and functions, statistics and probability, geometry and trigonometry, and discrete mathematics. The first three courses in the series provide a common core of broadly useful mathematics,…

Algebraic Trigonometry

ERIC Educational Resources Information Center

Vaninsky, Alexander

2011-01-01

This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos--satisfying an axiom sin[superscript 2] + cos[superscript 2] = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two…

Physics Data Booklet (Revised 1987).

ERIC Educational Resources Information Center

Alberta Dept. of Education, Edmonton.

This booklet was designed as a reference for teachers and students of physics on various types of data. Included are: (1) formulas for various constants involved in the study of gravity, electricity, magnetism, atomic physics, particles, and trigonometry; (2) a chart containing values of trigometric functions; (3) equations used in the study of…

Tour of a Simple Trigonometry Problem

ERIC Educational Resources Information Center

Poon, Kin-Keung

2012-01-01

This article focuses on a simple trigonometric problem that generates a strange phenomenon when different methods are applied to tackling it. A series of problem-solving activities are discussed, so that students can be alerted that the precision of diagrams is important when solving geometric problems. In addition, the problem-solving plan was…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12. Revised.

ERIC Educational Resources Information Center

Cranston School Dept., RI.

Behavioral objectives for grades 10 through 12 are specified for plane geometry, algebra, general mathematics, computer mathematics, slide rule mathematics, basic college mathematics, trigonometry, analytic geometry, calculus and probability. Most sections present material in terms of portions of a school year. At least one major objective is…

Calculus in High School--At What Cost?

ERIC Educational Resources Information Center

Sorge, D. H.; Wheatley, G. H.

1977-01-01

Evidence on the decline in preparation of entering calculus students and the relationship to high school preparation is presented, focusing on the trend toward the de-emphasis of trigonometry and analytic geometry in favor of calculus. Data on students' perception of the adequacy of their preparation are also presented. (Author/MN)

Using Technology to Promote Mathematical Discourse Concerning Women in Mathematics

ERIC Educational Resources Information Center

Phy, Lyn

2008-01-01

This paper discusses uses of technology to facilitate mathematical discourse concerning women in mathematics. Such a topic can be introduced in various traditional courses such as algebra, geometry, trigonometry, probability and statistics, or calculus, but it is not included in traditional textbooks. Through the ideas presented here, you can…

DIFFERENTIAL AND INTEGRAL CALCULUS. A TENTATIVE CURRICULUM GUIDE.

ERIC Educational Resources Information Center

BRANT, VINCENT; GERARDI, WILLIAM

A GUIDE FOR A 1-YEAR COURSE IN DIFFERENTIAL AND INTEGRAL CALCULUS PREREQUISITED KNOWLEDGE IN ALGEBRA, ANALYTIC TRIGONOMETRY, AND ELEMENTARY ANALYSIS. EACH ASSIGNMENT CONTAINED BOTH NEW AND REVIEW WORK TO REINFORCE THE NEW WORK. THERE WERE ELEVEN UNITS OF STUDY USING THE FOLLOWING FOUR BOOKS--"CALCULUS AND ANALYTIC GEOMETRY, THIRD…

Bridges with Trigonometry Equals Engineering Achievement

ERIC Educational Resources Information Center

Gathing, Ahmed L.

2011-01-01

Exemplary and fun technology education classes in high schools are always welcome. The author introduces bridge building to his ninth graders and other students who comprise the Introduction to Engineering and Technology course within the first two months of the fall semester. In Georgia, Introduction to Engineering and Technology is the first of…

An Analysis of Singaporean versus Indonesian Textbooks Based on Trigonometry Content

ERIC Educational Resources Information Center

Yang, Der-Ching; Sianturi, Iwan Andi

2017-01-01

Organization for Economic Co-operation and Development (OECD) (2016) released the results of Programme for International Student Assessment (PISA) 2015 and reported that the students' performance in mathematics of Singapore and Indonesia had significant differences. There is a strong relationship between textbooks used and mathematics performance…

Hermann-Bernoulli-Laplace-Hamilton-Runge-Lenz Vector.

ERIC Educational Resources Information Center

Subramanian, P. R.; And Others

1991-01-01

A way for students to refresh and use their knowledge in both mathematics and physics is presented. By the study of the properties of the "Runge-Lenz" vector the subjects of algebra, analytical geometry, calculus, classical mechanics, differential equations, matrices, quantum mechanics, trigonometry, and vector analysis can be reviewed. (KR)

The Trouble with Trig

ERIC Educational Resources Information Center

Galle, Gillian; Meredith, Dawn

2014-01-01

A few years ago we began to revamp our introductory physics course for life science students. We knew that this cohort would be less prepared and less adventurous mathematically than engineering, physical science, or mathematics majors. Moreover, from our own experience and the mathematics education literature, we knew that trigonometry would be…

The Mathematics of Sundials

ERIC Educational Resources Information Center

Vincent, Jill

2008-01-01

As early as 3500 years ago, shadows of sticks were used as a primitive instrument for indicating the passage of time through the day. The stick came to be called a "gnomon" or "one who knows." Early Babylonian obelisks were designed to determine noon. The development of trigonometry by Greek mathematicians meant that hour lines…

Spiral Growth in Plants: Models and Simulations

ERIC Educational Resources Information Center

Allen, Bradford D.

2004-01-01

The analysis and simulation of spiral growth in plants integrates algebra and trigonometry in a botanical setting. When the ideas presented here are used in a mathematics classroom/computer lab, students can better understand how basic assumptions about plant growth lead to the golden ratio and how the use of circular functions leads to accurate…

[North Carolina Gifted and Talented Minigrant Curriculum Projects: Two Microcomputer Projects].

ERIC Educational Resources Information Center

Parrish, Ronald; Baker, Reginald

Computer awareness and literacy programs for gifted and talented high school students were developed at two Washington City Schools (North Carolina). At Carteret High School, a variety of computer programs were purchased for biology and physics studies, trigonometry and algebra studies, aptitude and merit exam preparation, basic skills math…

Recalling Prerequisite Material in a Calculus II Course to Improve Student Success

ERIC Educational Resources Information Center

Mokry, Jeanette

2016-01-01

This article discusses preparation assignments used in a Calculus II course that cover material from prerequisite courses. Prior to learning new material, students work on problems outside of class involving concepts from algebra, trigonometry, and Calculus I. These problems are directly built upon in order to answer Calculus II questions,…

Trigonometry from a Different Angle

ERIC Educational Resources Information Center

Cavanagh, Michael

2008-01-01

The mathematics methodology subjects the author undertook in the early 1980s encouraged him to adopt a very expository style of teaching in which each new concept is introduced by its formal definition. The teacher should then explain a few carefully chosen examples for students to copy into their books, and then provide plenty of graded practice…

An Examination of High School Students' Online Engagement in Mathematics Problems

ERIC Educational Resources Information Center

Lim, Woong; Son, Ji-Won; Gregson, Susan; Kim, Jihye

2018-01-01

This article examines high school students' engagement in a set of trigonometry problems. Students completed this task independently in an online environment with access to Internet search engines, online textbooks, and YouTube videos. The findings imply that students have the resourcefulness to solve procedure-based mathematics problems in an…

A Subject Classification of Math Lab Activities from School Science and Mathematics 1974-1980.

ERIC Educational Resources Information Center

Grinstein, Louise S.

1982-01-01

Presented here is an index which indicates the title and location of each activity by volume and page numbers. The majority of items relate to arithmetic, elementary algebra, and plane geometry, but material also covers such topics as statistics, probability, trigonometry set theory, topology, and modern algebra. (MP)

University of Chicago School Mathematics Project 6-12 Curriculum. What Works Clearinghouse Intervention Report

ERIC Educational Resources Information Center

What Works Clearinghouse, 2011

2011-01-01

The "University of Chicago School Mathematics Project ("UCSMP") 6-12 Curriculum" is a series of yearlong courses--(1) Transition Mathematics; (2) Algebra; (3) Geometry; (4) Advanced Algebra; (5) Functions, Statistics, and Trigonometry; and (6) Precalculus and Discrete Mathematics--emphasizing problem solving, real-world applications, and the use…

Calculus of Elementary Functions, Part II. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This teacher's guide is for Part II of the course. It is designed to follow Part I of the text. The guide contains background information, suggested instructional…

Calculus of Elementary Functions, Part I. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part I, contains the first five chapters of the course and two appendices. Chapters included are: (1) Polynomial Functions; (2) The Derivative of a Polynomial…

Calculus of Elementary Functions, Part II. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part II, contains material designed to follow Part I. Chapters included in this text are: (6) Derivatives of Exponential and Related Functions; (7) Area and…

STUDY OF VARIABLES ASSOCIATED WITH FINAL GRADES IN MATHEMATICS COURSES.

ERIC Educational Resources Information Center

DAVIS, ELTON C.; RISSER, JOHN J.

THIS STUDY WAS CONDUCTED IN ORDER TO DETERMINE THE RELATIVE VALUE OF PREVIOUS GRADES IN MATHEMATICS COURSES, THE OVERALL HIGH SCHOOL GRADE POINT AVERAGE, AND THE PLACEMENT TEST IN MATHEMATICS DEVELOPED AT THE COLLEGE AS PREDICTORS OF ACHIEVEMENT IN INTRODUCTORY AND INTERMEDIATE ALGEBRA, IN COLLEGE ALGEBRA, IN TRIGONOMETRY, AND IN ANALYTIC GEOMETRY…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12, Levels 87-112.

ERIC Educational Resources Information Center

Rogers, Arnold R., Ed.; And Others

Behavioral objectives for geometry, algebra, computer mathematics, trigonometry, analytic geometry, calculus, and probability are specified for grades 10 through 12. General objectives are stated for major areas under each topic and are followed by a list of specific objectives for that area. This work was prepared under an ESEA Title III…

Sighting the International Space Station

ERIC Educational Resources Information Center

Teets, Donald

2008-01-01

This article shows how to use six parameters describing the International Space Station's orbit to predict when and in what part of the sky observers can look for the station as it passes over their location. The method requires only a good background in trigonometry and some familiarity with elementary vector and matrix operations. An included…

An asymptotical machine

NASA Astrophysics Data System (ADS)

Cristallini, Achille

2016-07-01

A new and intriguing machine may be obtained replacing the moving pulley of a gun tackle with a fixed point in the rope. Its most important feature is the asymptotic efficiency. Here we obtain a satisfactory description of this machine by means of vector calculus and elementary trigonometry. The mathematical model has been compared with experimental data and briefly discussed.

Computational trigonometry

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gustafson, K.

1994-12-31

By means of the author`s earlier theory of antieigenvalues and antieigenvectors, a new computational approach to iterative methods is presented. This enables an explicit trigonometric understanding of iterative convergence and provides new insights into the sharpness of error bounds. Direct applications to Gradient descent, Conjugate gradient, GCR(k), Orthomin, CGN, GMRES, CGS, and other matrix iterative schemes will be given.

Evolving Polygons and Spreadsheets: Connecting Mathematics across Grade Levels in Teacher Education

ERIC Educational Resources Information Center

Abramovich, Sergei; Brouwer, Peter

2009-01-01

This paper was prepared in response to the Conference Board of Mathematical Sciences recommendations for the preparation of secondary teachers. It shows how using trigonometry as a conceptual tool in spreadsheet-based applications enables one to develop mathematical understanding in the context of constructing geometric representations of unit…

Instructional Decision Making and Agency of Community College Mathematics Faculty

ERIC Educational Resources Information Center

Lande, Elaine; Mesa, Vilma

2016-01-01

We investigate the rationale for instructional decisions proposed by two groups of community college mathematics faculty (full-time and part-time), as they discussed animations of trigonometry classes that breached several classroom norms. Although both groups of faculty justify their decisions in similar ways, the way in which they talk differs.…

Mathematics education graduate students' understanding of trigonometric ratios

NASA Astrophysics Data System (ADS)

Yiǧit Koyunkaya, Melike

2016-10-01

This study describes mathematics education graduate students' understanding of relationships between sine and cosine of two base angles in a right triangle. To explore students' understanding of these relationships, an elaboration of Skemp's views of instrumental and relational understanding using Tall and Vinner's concept image and concept definition was developed. Nine students volunteered to complete three paper and pencil tasks designed to elicit evidence of understanding and three students among these nine students volunteered for semi-structured interviews. As a result of fine-grained analysis of the students' responses to the tasks, the evidence of concept image and concept definition as well as instrumental and relational understanding of trigonometric ratios was found. The unit circle and a right triangle were identified as students' concept images, and the mnemonic was determined as their concept definition for trigonometry, specifically for trigonometric ratios. It is also suggested that students had instrumental understanding of trigonometric ratios while they were less flexible to act on trigonometric ratio tasks and had limited relational understanding. Additionally, the results indicate that graduate students' understanding of the concept of angle mediated their understanding of trigonometry, specifically trigonometric ratios.

Astronomy and Mathematics Education

NASA Astrophysics Data System (ADS)

Ros, Rosa M.

There are many European countries where Astronomy does not appear as a specific course on the secondary school. In these cases Astronomy content can be introduced by means of other subjects. There are some astronomical topics within the subject of Physics but this talk concerns introducing Astronomy in Mathematics classes. Teaching Astronomy through Mathematics would result in more exposure than through Physics as Mathematics is more prevalent in the curriculum. Generally it is not easy to motivate students in Mathematics but they are motivated to find out more about the universe and Astronomy current events than appears in the media. This situation can be an excellent introduction to several mathematics topics. The teachers in secondary and high school can use this idea in order to present more attractive mathematics courses. In particular some different examples will be offered regarding * Angles and spherical coordinates considering star traces * Logarithms and visual magnitudes * Plane trigonometry related orbital movements * Spherical trigonometry in connection with ecliptic obliquity * Conic curves related to sundial at several latitudes Some students do not enjoy studying Mathematics but they can be attracted by practical situations using Applied Mathematics: Astronomy is always very attractive to teenagers.

Software for Training in Pre-College Mathematics

NASA Technical Reports Server (NTRS)

Shelton, Robert O.; Moebes, Travis A.; VanAlstine, Scot

2003-01-01

The Intelligent Math Tutor (IMT) is a computer program for training students in pre-college and college-level mathematics courses, including fundamentals, intermediate algebra, college algebra, and trigonometry. The IMT can be executed on a server computer for access by students via the Internet; alternatively, it can be executed on students computers equipped with compact- disk/read-only-memory (CD-ROM) drives. The IMT provides interactive exercises, assessment, tracking, and an on-line graphing calculator with algebraic-manipulation capabilities. The IMT provides an innovative combination of content, delivery mechanism, and artificial intelligence. Careful organization and presentation of the content make it possible to provide intelligent feedback to the student based on performance on exercises and tests. The tracking and feedback mechanisms are implemented within the capabilities of a commercial off-the-shelf development software tool and are written in the Unified Modeling Language to maximize reuse and minimize development cost. The graphical calculator is a standard feature of most college and pre-college algebra and trigonometry courses. Placing this functionality in a Java applet decreases the cost, provides greater capabilities, and provides an opportunity to integrate the calculator with the lessons.

Improving Student Success in Calculus I Using a Co-Requisite Calculus I Lab

ERIC Educational Resources Information Center

Vestal, Sharon Schaffer; Brandenburger, Thomas; Furth, Alfred

2015-01-01

This paper describes how one university mathematics department was able to improve student success in Calculus I by requiring a co-requisite lab for certain groups of students. The groups of students required to take the co-requisite lab were identified by analyzing student data, including Math ACT scores, ACT Compass Trigonometry scores, and…

al-Biruni, Abu Raihan (973-1048)

NASA Astrophysics Data System (ADS)

Murdin, P.

2000-11-01

Born in Kheva near Ural, present-day Uzbekhistan, al-Biruni was a polymath and traveler (to India), making contributions in mathematics, geography and geology, natural history, calendars and astronomy. His book Qanun-i Masoodi, which he dedicated to his patron Sultan Masood, discusses astronomy, trigonometry, solar, lunar and planetary motions, including the question whether the Earth rotates or ...

Triangles with Integer Side Lengths and Rational Internal Radius P and External Radius R

ERIC Educational Resources Information Center

Zelator, Konstantine

2005-01-01

This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…

Notes for Applied Mathematics in Trigonometry and Earth Geometry/Navigation

ERIC Educational Resources Information Center

Faulkner, Peter

2004-01-01

As time has progressed, the role of applied mathematics has become increasingly important. Indeed there are now more students enrolled in applied mathematics courses in senior high schools and colleges than in pure mathematics. Such courses become more relevant both to the student and to future employers, if the same constants and equations that…

The Effect of the Math Emporium Instructional Method on Students' Performance in College Algebra

ERIC Educational Resources Information Center

Cousins-Cooper, Kathy; Staley, Katrina N.; Kim, Seongtae; Luke, Nicholas S.

2017-01-01

This study aims to investigate the effectiveness of the Emporium instructional method in a course of college algebra and trigonometry by comparing to the traditional lecture method. The math emporium method is a nontraditional instructional method of learning math that has been implemented at several universities with much success and has been…

Exploring Magnetic Fields with a Compass

ERIC Educational Resources Information Center

Lunk, Brandon; Beichner, Robert

2011-01-01

A compass is an excellent classroom tool for the exploration of magnetic fields. Any student can tell you that a compass is used to determine which direction is north, but when paired with some basic trigonometry, the compass can be used to actually measure the strength of the magnetic field due to a nearby magnet or current-carrying wire. In this…

Individualized Math Problems in Trigonometry. Oregon Vo-Tech Mathematics Problem Sets.

ERIC Educational Resources Information Center

Cosler, Norma, Ed.

This is one of eighteen sets of individualized mathematics problems developed by the Oregon Vo-Tech Math Project. Each of these problem packages is organized around a mathematical topic and contains problems related to diverse vocations. Solutions are provided for all problems. Problems in this volume require the use of trigonometric and inverse…

Elements of Mathematics, Book O: Intuitive Background. Chapter 14, Geometry: Similitudes, Coordinates, and Trigonometry.

ERIC Educational Resources Information Center

Exner, Robert; And Others

The sixteen chapters of this book provide the core material for the Elements of Mathematics Program, a secondary sequence developed for highly motivated students with strong verbal abilities. The sequence is based on a functional-relational approach to mathematics teaching, and emphasizes teaching by analysis of real-life situations. This text is…

Biomedical Mathematics, Unit I: Measurement, Linear Functions and Dimensional Algebra. Student Text. Revised Version, 1975.

ERIC Educational Resources Information Center

Biomedical Interdisciplinary Curriculum Project, Berkeley, CA.

This text presents lessons relating specific mathematical concepts to the ideas, skills, and tasks pertinent to the health care field. Among other concepts covered are linear functions, vectors, trigonometry, and statistics. Many of the lessons use data acquired during science experiments as the basis for exercises in mathematics. Lessons present…

The Development of an Individualized Instructional Program in Beginning College Mathematics Utilizing Computer Based Resource Units. Final Report.

ERIC Educational Resources Information Center

Rockhill, Theron D.

Reported is an attempt to develop and evaluate an individualized instructional program in pre-calculus college mathematics. Four computer based resource units were developed in the areas of set theory, relations and function, algebra, trigonometry, and analytic geometry. Objectives were determined by experienced calculus teachers, and…

Using Manipulatives to Support an Embodied Approach to Learning Trigonometry in a South African School: A Case Study

ERIC Educational Resources Information Center

Brijlall, D.; Niranjan, C.

2015-01-01

Multiple Intelligence Theory suggests that individuals perceive knowledge in eight different ways. This article reports on a study that explored the role of manipulatives in the teaching and learning of trigonometric ratios in grade 10. The approach attempts in addressing three domains of the Multiple Intelligence Theory (linguistic/verbal…

A Conceptual Analysis of the Knowledge of Prospective Mathematics Teachers about Degree and Radian

ERIC Educational Resources Information Center

Tuna, Abdulkadir

2013-01-01

This study examined the knowledge levels of prospective mathematics teachers about the concepts of degree and radian, which are among the angle measuring units that constitute the basis of trigonometry, and the relationships between those concepts. The study group consisted of 93 prospective mathematics teachers attending a state university in…

Cooperative Learning in the Advanced Algebra and Trigonometry Mathematics High School Classroom

ERIC Educational Resources Information Center

Jozsa, Alison

2017-01-01

Over the past three decades, researchers have found cooperative learning to have positive effects on student achievement in various subject areas and levels in education. However, there are limited studies on the impact of cooperative learning on student achievement in the area of high school mathematics. This study examined the impact of…

Teaching a Concept with GeoGebra: Periodicity of Trigonometric Functions

ERIC Educational Resources Information Center

Kepceoglu, Ibrahim; Yavuz, llyas

2016-01-01

Being one of the major subjects in high school mathematics curriculum, trigonometry links algebraic, geometric and graphical reasoning. The aim of this study is to investigate the effect of GeoGebra in the teaching of the concept of the periodicity of trigonometric functions. In this study, it is investigated how effective is the dynamic…

Tile Patterns with LOGO--Part III: Tile Patterns from Mult Tiles Using Logo.

ERIC Educational Resources Information Center

Clason, Robert G.

1991-01-01

A mult tile is a set of polygons each of which can be dissected into smaller polygons similar to the original set of polygons. Using a recursive LOGO method that requires solutions to various geometry and trigonometry problems, dissections of mult tiles are carried out repeatedly to produce tile patterns. (MDH)

Unit Circles and Inverse Trigonometric Functions

ERIC Educational Resources Information Center

Barrera, Azael

2014-01-01

Historical accounts of trigonometry refer to the works of many Indian and Arab astronomers on the origin of the trigonometric functions as we know them now, in particular Abu al-Wafa (ca. 980 CE), who determined and named all known trigonometric functions from segments constructed on a regular circle and later on a unit circle (Moussa 2011;…

Computing relative plate velocities: a primer

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bevis, M.

1987-08-01

Standard models of present-day plate motions are framed in terms of rates and poles of rotation, in accordance with the well-known theorem due to Euler. This article shows how computation of relative plate velocities from such models can be viewed as a simple problem in spherical trigonometry. A FORTRAN subroutine is provided to perform the necessary computations.

Discovering Trigonometric Relationships Implied by the Law of Sines and the Law of Cosines

ERIC Educational Resources Information Center

Skurnick, Ronald; Javadi, Mohammad

2006-01-01

The Law of Sines and The Law of Cosines are of paramount importance in the field of trigonometry because these two theorems establish relationships satisfied by the three sides and the three angles of any triangle. In this article, the authors use these two laws to discover a host of other trigonometric relationships that exist within any…

Implications of a Comparative Study for Mathematics Education in the English Education System

ERIC Educational Resources Information Center

Delice, Ali; Roper, Tom

2006-01-01

This paper reports upon particular aspects of a study carried out by Delice in 2003, the main aim of which was to compare the performance of students in the 16-19 age group from Turkey and England on trigonometry of "A-level standard" and then to compare the curriculum and assessment provision in each country to seek possible…

Teaching Guide and Problem Supplement. A Publication of the Exemplary Project Problem Solving Computer Style 1969-1970.

ERIC Educational Resources Information Center

New Orleans Public Schools, LA.

Secondary school teachers incorporating the use of a computer in algebra, trigonometry, advanced mathematics, chemistry, or physics classes are the individuals for whom this book is intended. The content included in it is designed to aid the learning of programing techniques and basic scientific or mathematical principles, and to offer some…

Poles, Parking Lots, and Mount Piton: Classroom Activities that Combine Astronomy, History, and Mathematics

ERIC Educational Resources Information Center

Madden, Sean P.; Comstock, Jocelyn M.; Downing, James P.

2006-01-01

This article describes how a series of lessons might be used to allow students to discover the size of the Earth, the distance to the Moon, the size of the Moon, and the altitude of Mount Piton on the Moon. Measurement with a sextant, principles of geometry and trigonometry, and historically important scientists and mathematicians are discussed.

Mathematics for the Workplace. Applications from Machine Tool Technology (Michelin Tire Corporation). A Teacher's Guide.

ERIC Educational Resources Information Center

Wallace, Johnny M.; Stewart, Grover

This module presents a real-world context in which mathematics skills (geometry and trigonometry) are used as part of a daily routine. The context is the machine tool technology field, and the module aims to help students develop the ability to analyze diagrams in order to make mathematical computations. The modules, which features applications…

A genotype probability index for multiple alleles and haplotypes.

PubMed

Percy, A; Kinghorn, B P

2005-12-01

We use linear algebra to calculate an index of information content in genotype probabilities which has previously been calculated using trigonometry. The new method can be generalized allowing the index to be calculated for loci with more than two alleles. Applications of this index include its use in genotyping strategies, strategies to manage genetic disorders and in estimation of genotype effects.

Math Readiness and Preparation for Competitive College Majors and Careers: The Case of Black Students.

ERIC Educational Resources Information Center

Thomas, Gail E.

This study examines factors that determine the enrollment of black students in the high school math courses (i.e., advanced algebra, trigonometry, calculus) that are necessary for competitive college and major field access. The data are from a local college survey of juniors and seniors who were enrolled in eight (8) local public and private…

An Exploration of College Students' Problem Solving Behaviors While Verifying Trigonometric Identities: A Mixed Methods Case Study

ERIC Educational Resources Information Center

Wescoatt, Benjamin Mark

2013-01-01

Topics in trigonometry have not been well-studied, especially with college-level students. Thus, despite providing a venue for important concepts such as notions of proof and algebraic skill, the process of verifying trigonometric identities, or VTI, has not been thoroughly explored. This study attempts to remedy this gap in the literature by…

Learning to Teach High School Mathematics: Patterns of Growth in Understanding Right Triangle Trigonometry during Lesson Plan Study

ERIC Educational Resources Information Center

Cavey, Laurie O.; Berenson, Sarah B.

2005-01-01

"Lesson plan study" (LPS), adapted from the Japanese Lesson Study method of professional development, is a sequence of activities designed to engage prospective teachers in broadening and deepening their understanding of school mathematics and teaching strategies. LPS occurs over 5 weeks on the same lesson topic and includes four opportunities to…

Quaternions in computer vision and robotics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Pervin, E.; Webb, J.A.

1982-01-01

Computer vision and robotics suffer from not having good tools for manipulating three-dimensional objects. Vectors, coordinate geometry, and trigonometry all have deficiencies. Quaternions can be used to solve many of these problems. Many properties of quaternions that are relevant to computer vision and robotics are developed. Examples are given showing how quaternions can be used to simplify derivations in computer vision and robotics.

Examining the Effects of Gender, Poverty, Attendance, and Ethnicity on Algebra, Geometry, and Trigonometry Performance in a Public High School

ERIC Educational Resources Information Center

Shafiq, Hasan

2013-01-01

Over the last few decades school accountability for student performance has become an issue at the forefront of education. The federal No Child Left Behind Act of 2001 (NCLB) and various regulations by individual states have set standards for student performance at both the district and individual public and charter school levels, and certain…

Space Mathematics: A Resource for Secondary School Teachers

NASA Technical Reports Server (NTRS)

Kastner, Bernice

1985-01-01

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

Evaluation of Markov-Decision Model for Instructional Sequence Optimization. Semi-Annual Technical Report for the period 1 July-31 December 1975. Technical Report No. 76.

ERIC Educational Resources Information Center

Wollmer, Richard D.; Bond, Nicholas A.

Two computer-assisted instruction programs were written in electronics and trigonometry to test the Wollmer Markov Model for optimizing hierarchial learning; calibration samples totalling 110 students completed these programs. Since the model postulated that transfer effects would be a function of the amount of practice, half of the students were…

Collinear collision chemistry. II. Energy disposition in reactive collisions

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mahan, B.H.

1974-06-01

A model describing the mechanics of collinear atom-diatom collisions and previously reported by the author is extended to describe reactive collisions. The model indicates the effects of such factors as the mass distribution and potential energy barriers and wells on the reaction probability and on the distribution of energy among the modes of motion of the products. Simple geometry and trigonometry are sufficient to solve the model.

Mobile Gait Analysis System for Lower Limb Amputee High-Level Activity Rehabilitation

DTIC Science & Technology

2013-09-01

The direction of gravity can be used, along with trigonometry, to determine the pitch and roll orientations of the IMU . We are interested in the...are represented using direction cosine matrices so pitch and roll rotations can be isolated while rotations about the gravity vector are ignored...three signals from the gyroscope in the IMU frame and any drift associated with the gyroscope, and . An estimate of roll and pitch, and

Study Modules for Calculus-Based General Physics. [Includes Modules 1 and 2: Dimensions and Vector Addition; Rectilinear Motion; plus a Trigonometry and Calculus Review].

ERIC Educational Resources Information Center

Fuller, Robert G., Ed.; And Others

This is part of a series of 42 Calculus Based Physics (CBP) modules totaling about 1,000 pages. The modules include study guides, practice tests, and mastery tests for a full-year individualized course in calculus-based physics based on the Personalized System of Instruction (PSI). The units are not intended to be used without outside materials;…

A Neuroergonomic Quasi-Experiment: Predictors of Situation Awareness and Display Usability with USAF Pilots while Performing Complex Tasks

DTIC Science & Technology

2015-02-01

adverse reactions possibly associated with electrode placement, such as urticarial (hives) or contact dermatitis . In such an event, the medical... contact (airplane’s wheels time of impact with the runway). The human brain uses internal trigonometry and calculus to solve the Time of arrival...and regulations, and that if you desire further information you may contact the base legal office (ASC/JA, 257-6143 for Wright-Patterson AFB). In

The Effectiveness of Problem-Based Learning Approach Based on Multiple Intelligences in Terms of Student’s Achievement, Mathematical Connection Ability, and Self-Esteem

NASA Astrophysics Data System (ADS)

Kartikasari, A.; Widjajanti, D. B.

2017-02-01

The aim of this study is to explore the effectiveness of learning approach using problem-based learning based on multiple intelligences in developing student’s achievement, mathematical connection ability, and self-esteem. This study is experimental research with research sample was 30 of Grade X students of MIA III MAN Yogyakarta III. Learning materials that were implemented consisting of trigonometry and geometry. For the purpose of this study, researchers designed an achievement test made up of 44 multiple choice questions with respectively 24 questions on the concept of trigonometry and 20 questions for geometry. The researcher also designed a connection mathematical test and self-esteem questionnaire that consisted of 7 essay questions on mathematical connection test and 30 items of self-esteem questionnaire. The learning approach said that to be effective if the proportion of students who achieved KKM on achievement test, the proportion of students who achieved a minimum score of high category on the results of both mathematical connection test and self-esteem questionnaire were greater than or equal to 70%. Based on the hypothesis testing at the significance level of 5%, it can be concluded that the learning approach using problem-based learning based on multiple intelligences was effective in terms of student’s achievement, mathematical connection ability, and self-esteem.

Sines and Cosines. Part 2 of 3

NASA Technical Reports Server (NTRS)

Apostol, Tom M. (Editor)

1993-01-01

The Law of Sines and the Law of Cosines are introduced and demonstrated in this 'Project Mathematics' series video using both film footage and computer animation. This video deals primarily with the mathematical field of Trigonometry and explains how these laws were developed and their applications. One significant use is geographical and geological surveying. This includes both the triangulation method and the spirit leveling method. With these methods, it is shown how the height of the tallest mountain in the world, Mt. Everest, was determined.

The myth of the 90 degrees-angle intramuscular injection.

PubMed

Katsma, D L; Katsma, R

2000-01-01

This article shows that the textbook 90 degrees-angle requirement for intramuscular injections is unrealistic. Trigonometry demonstrates that an injection given at 72 degrees reaches 95% of the depth of an injection given at 90 degrees. This relation between needle angle and needle depth, previous research into the kinematics of hand motion during an intramuscular injection, and other practical considerations support the proposal for a new, relaxed standard: Intramuscular injections administered at a comfortable angle between 72 degrees and 90 degrees.

Tour of a simple trigonometry problem

NASA Astrophysics Data System (ADS)

Poon, Kin-Keung

2012-06-01

This article focuses on a simple trigonometric problem that generates a strange phenomenon when different methods are applied to tackling it. A series of problem-solving activities are discussed, so that students can be alerted that the precision of diagrams is important when solving geometric problems. In addition, the problem-solving plan was implemented in a high school and the results indicated that students are relatively weak in problem-solving abilities but they understand and appreciate the thinking process in different stages and steps of the activities.

Handbook of applied mathematics for engineers and scientists

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kurtz, M.

1991-12-31

This book is intended to be reference for applications of mathematics in a wide range of topics of interest to engineers and scientists. An unusual feature of this book is that it covers a large number of topics from elementary algebra, trigonometry, and calculus to computer graphics and cybernetics. The level of mathematics covers high school through about the junior level of an engineering curriculum in a major univeristy. Throughout, the emphasis is on applications of mathematics rather than on rigorous proofs.

Determining the Altitude of Iridium Flares

NASA Technical Reports Server (NTRS)

Foster, James; Owe, Manfred

1999-01-01

Iridium flares have nothing to do with the element iridium. Iridium is also the name of a telecommunications company that has been launching satellites into low orbits around the Earth. These satellites are being used for a new type of wireless phone and paging service. Flares have been observed coming from these satellites. These flares have the potential, especially when the full fleet of satellites is in orbit, to disrupt astronomical observations. The paper reviews using simple trigonometry how to calculate the altitude of one of these satellites.

Easing The Calculation Of Bolt-Circle Coordinates

NASA Technical Reports Server (NTRS)

Burley, Richard K.

1995-01-01

Bolt Circle Calculation (BOLT-CALC) computer program used to reduce significant time consumed in manually computing trigonometry of rectangular Cartesian coordinates of holes in bolt circle as shown on blueprint or drawing. Eliminates risk of computational errors, particularly in cases involving many holes or in cases in which coordinates expressed to many significant digits. Program assists in many practical situations arising in machine shops. Written in BASIC. Also successfully compiled and implemented by use of Microsoft's QuickBasic v4.0.

Positive academic emotions moderate the relationship between self-regulation and academic achievement.

PubMed

Villavicencio, Felicidad T; Bernardo, Allan B I

2013-06-01

Research has shown how academic emotions are related to achievement and to cognitive/motivational variables that promote achievement. Mediated models have been proposed to account for the relationships among academic emotions, cognitive/motivational variables, and achievement, and research has supported such mediated models, particularly with negative emotions. The study tested the hypotheses: (1) self-regulation and the positive academic emotions of enjoyment and pride are positive predictors of achievement; and (2) enjoyment and pride both moderate the relationship between self-regulation and achievement. Participants were 1,345 students enrolled in various trigonometry classes in one university. Participants answered the Academic Emotions Questionnaire-Math (Pekrun, Goetz, & Frenzel, 2005) and a self-regulation scale (Pintrich, Smith, Garcia, & McKeachie, 1991) halfway through their trigonometry class. The students' final grades in the course were regressed to self-regulation, positive emotions, and the interaction terms to test the moderation effects. Enjoyment and pride were both positive predictors of grades; more importantly, both moderated the relationship between self-regulation and grades. For students who report higher levels of both positive emotions, self-regulation was positively associated with grades. However, for those who report lower levels of pride, self-regulation was not related to grades; and, for those who reported lower levels of enjoyment, self-regulation was negatively related to grades. The results are discussed in terms of how positive emotions indicate positive appraisals of task/outcome value, and thus enhance the positive links between cognitive/motivational variables and learning. ©2012 The British Psychological Society.

Flat Earth theory: an exercise in critical thinking

NASA Astrophysics Data System (ADS)

Břízová, Leontýna; Gerbec, Kelsey; Šauer, Jiří; Šlégr, Jan

2018-07-01

In this paper we present a critical analysis of some of the arguments of flat Earth theory, and we also try to show that this analysis and refutation of these false claims can be a useful exercise in critical thinking that is so much needed today. This article can also make it easier for teachers who are exposed to some of the arguments of flat Earth theory by their students. Some arguments of this theory are completely senseless, and some can simply be disproved by trigonometry or basic physical laws.

Broad source fringe formation with a Fresnel biprism and a Mach-Zehnder interferometer.

PubMed

Leon, S C

1987-12-15

A biprism is used to combine identical spatially incoherent wavefronts that have been split by an amplitude splitting interferometer such as the Mach-Zehnder. The performance of this composite interferometer is evaluated by tracing the chief ray through parallel optical systems using Snell's law and trigonometry. Fringes formed in spatially incoherent light with this optical system are compared with those formed using the Mach-Zehnder and grating interferometers. It is shown that the combination can exhibit extended source fringe formation capability greatly exceeding that of the Mach-Zehnder interferometer.

The properties of Q-deformed hyperbolic and trigonometric functions in quantum deformation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Deta, U. A., E-mail: utamaalan@yahoo.co.id, E-mail: utamadeta@unesa.ac.id; Suparmi

2015-09-30

Quantum deformation has been studied due to its relation with applications in nuclear physics, conformal field theory, and statistical-quantum theory. The q-deformation of hyperbolic function was introduced by Arai. The application of q-deformed functions has been widely used in quantum mechanics. The properties of this two kinds of system explained in this paper including their derivative. The graph of q-deformed functions presented using Matlab. The special case is given for modified Poschl-Teller plus q-deformed Scarf II trigonometry potentials.

Maximum range of a projectile launched from a height h: a non-calculus treatment

NASA Astrophysics Data System (ADS)

Ganci, S.; Lagomarsino, D.

2014-07-01

The classical example of problem solving, maximizing the range of a projectile launched from height h with velocity v over the ground level, has received various solutions. In some of these, one can find the maximization of the range R by differentiating R as a function of an independent variable or through the implicit differentiation in Cartesian or polar coordinates. In other papers, various elegant non-calculus solutions can be found. In this paper, this problem is revisited on the basis of the elementary analytical geometry and the trigonometry only.

Orientation of doubly rotated quartz plates.

PubMed

Sherman, J R

1989-01-01

A derivation from classical spherical trigonometry of equations to compute the orientation of doubly-rotated quartz blanks from Bragg X-ray data is discussed. These are usually derived by compact and efficient vector methods, which are reviewed briefly. They are solved by generating a quadratic equation with numerical coefficients. Two methods exist for performing the computation from measurements against two planes: a direct solution by a quadratic equation and a process of convergent iteration. Both have a spurious solution. Measurement against three lattice planes yields a set of three linear equations the solution of which is an unambiguous result.

A Novel Approach to Anharmonicity for a Wealth of Applications in Nonlinear Science Technologies

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hanusse, Patrick

2011-04-19

We present a new theory of the anharmonicity of nonlinear oscillations that are exhibited by many physical systems. New physical quantities are introduced that describe the departure from linear or harmonic behavior and as far as extremely anharmonic situations. In order to solve the nonlinear phase equation, the key notion of our theory, which controls the anharmonic behavior, a new and fascinating nonlinear trigonometry is designed. These results provide a general and accurate yet compact description of such signals, by far better than the Fourier description, both quantitatively and qualitatively and will benefit many application fields.

Cosmic strings: Gravitation without local curvature

DOE Office of Scientific and Technical Information (OSTI.GOV)

Helliwell, T.M.; Konkowski, D.A.

1987-05-01

Cosmic strings are very long, thin structures which might stretch over vast reaches of the universe. If they exist, they would have been formed during phase transitions in the very early universe. The space-time surrounding a straight cosmic string is flat but nontrivial: A two-dimensional spatial section is a cone rather than a plane. This feature leads to unique gravitational effects. The flatness of the cone means that many of the gravitational effects can be understood with no mathematics beyond trigonometry. This includes the observational predictions of the double imaging of quasars and the truncation of the images of galaxies.

Mathematical misconception in calculus 1: Identification and gender difference

NASA Astrophysics Data System (ADS)

Nassir, Asyura Abd; Abdullah, Nur Hidayah Masni; Ahmad, Salimah; Tarmuji, Nor Habibah; Idris, Aminatul Solehah

2017-08-01

A few years of experience of teaching mathematics make us notice that the same types of mistakes are done repeatedly by students. This paper presents an insight into categories of mistakes, how male and female students differ in terms of mistakes that are commonly done and the ability of the students to identify the mistakes. Sample of mistakes were taken from Calculus 1 final exam answer scripts, then it was listed and analyzed. Data analysis revealed that students' misconceptions fall into four categories. The first category is misunderstanding the meaning of brackets, followed by misconception of basic mathematics rules, misconception in notation and misconception in properties of trigonometry. A mistake identification test which consists of ten false mathematical statements was designed based on the mistake done by the previous batch of students that covered topics algebra, trigonometry, index, limit, differentiation and integration. Then, the test was given to students who enrolled in Calculus I course. Respondents of this study were randomly selected among two hundreds engineering students. Data obtained were analyzed using basic descriptive analysis and Chi Square test to capture gender differences in the mistake done for each category. Findings indicate that thirty five percent of the students have the ability to identify the mistakes and make a proper correction for at most two statements. Thirty one percent of the students are able to identify the mistakes but unable to make proper correction. Twenty five percent of the students failed to identify the mistakes in six out of ten false statements. Female students' misconception is more likely in basic mathematics rules compared to male. The findings of this study could serve as baseline information to be stressed in improving teaching and learning mathematics.

Comparison of postoperative vault height predictability using white-to-white or sulcus diameter-based sizing for the visian implantable collamer lens.

PubMed

Reinstein, Dan Z; Lovisolo, Carlo F; Archer, Timothy J; Gobbe, Marine

2013-01-01

To compare vault height predictability of Implantable Collamer Lens (ICL; Staar Surgical) sizing using a sulcus diameter-based formula or the manufacturer-recommended white-to-white-based method. In 50 myopic eyes, ICL size was calculated using both a formula including sulcus diameter and the traditional formula based on white-to-white diameter. Sulcus diameter was measured using Artemis 2 very high-frequency (VHF) digital ultrasound (ArcScan Inc). Implantation was based on the sulcus diameter derived size. Actual postoperative vault height achieved was measured by VHF digital ultrasound scanning. Circle segment trigonometry was used to calculate the vault height that would have resulted had lens sizing been based on the white-to-white formula. The same lens size would have been used in 60% of eyes, a smaller lens would have been used in 34% of eyes and a larger lens in 6% of eyes had lens sizing been based on the white-to-white formula. Mean vault for eyes with lenses sized using the sulcus diameter formula was 0.37±0.16 mm (range: 0.08 to 0.92 mm), with 2% <0.09 mm, the recognized low-vault height for risk of cataract. Circle segment trigonometry predicted that the vault height would have been 0.24±0.28 mm (range: -0.31 to 0.92 mm), with 26% <0.09 mm had lens sizing been based on the white-to-white formula. Significantly better predictability of postoperative vault height was achieved by including sulcus diameter into the ICL sizing formula compared with using the traditional white-to-white-based formula. Copyright 2013, SLACK Incorporated.

Measurement of whole tire profile

NASA Astrophysics Data System (ADS)

Yang, Yongyue; Jiao, Wenguang

2010-08-01

In this paper, a precision measuring device is developed for obtaining characteristic curve of tire profile and its geometric parameters. It consists of a laser displacement measurement unit, a closed-loop precision two-dimensional coordinate table, a step motor control system and a fast data acquisition and analysis system. Based on the laser trigonometry, a data map of tire profile and coordinate values of all points can be obtained through corresponding data transformation. This device has a compact structure, a convenient control, a simple hardware circuit design and a high measurement precision. Experimental results indicate that measurement precision can meet the customer accuracy requirement of +/-0.02 mm.

Diagonalization of the symmetrized discrete i th right shift operator

NASA Astrophysics Data System (ADS)

Fuentes, Marc

2007-01-01

In this paper, we consider the symmetric part of the so-called ith right shift operator. We determine its eigenvalues as also the associated eigenvectors in a complete and closed form. The proposed proof is elementary, using only basical skills such as Trigonometry, Arithmetic and Linear algebra. The first section is devoted to the introduction of the tackled problem. Second and third parts contain almost all the ?technical? stuff of the proofE Afterwards, we continue with the end of the proof, provide a graphical illustration of the results, as well as an application on the polyhedral ?sandwiching? of a special compact of arising in Signal theory.

On the helical arrangements of protein molecules.

PubMed

Dauter, Zbigniew; Jaskolski, Mariusz

2018-03-01

Helical structures are prevalent in biology. In the PDB, there are many examples where protein molecules are helically arranged, not only according to strict crystallographic screw axes but also according to approximate noncrystallographic screws. The preponderance of such screws is rather striking as helical arrangements in crystals must preserve an integer number of subunits per turn, while intuition and simple packing arguments would seem to favor fractional helices. The article provides insights into such questions, based on stereochemistry, trigonometry, and topology, and illustrates the findings with concrete PDB structures. Updated statistics of Sohncke space groups in the PDB are also presented. © 2017 The Protein Society.

DOE Fundamentals Handbook: Mathematics, Volume 1

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

Techniques for plotting shadow patterns

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bainbridge, D.A.

1982-02-01

Basic approaches for plotting shadow patterns (summer or winter) are discussed, illustrated, and compared. The solar simulator technique uses floodlights or a moveable table to mimic the sun's path over a model of the building being studied. The drawback is that, for large developments, very small models would have to be built. Graphic solutions (2 types) are described in which: (1) sun angles are used to calculate shadow patterns using trigonometry and (2) drawings are made and shadows are calculated. Examples are given for a house on level ground and on sloping ground. Calculations of shade density are also illustrated.more » 8 references. (MJJ)« less

DOE Fundamentals Handbook: Mathematics, Volume 2

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

The perception of geometrical structure from congruence

NASA Technical Reports Server (NTRS)

Lappin, Joseph S.; Wason, Thomas D.

1989-01-01

The principle function of vision is to measure the environment. As demonstrated by the coordination of motor actions with the positions and trajectories of moving objects in cluttered environments and by rapid recognition of solid objects in varying contexts from changing perspectives, vision provides real-time information about the geometrical structure and location of environmental objects and events. The geometric information provided by 2-D spatial displays is examined. It is proposed that the geometry of this information is best understood not within the traditional framework of perspective trigonometry, but in terms of the structure of qualitative relations defined by congruences among intrinsic geometric relations in images of surfaces. The basic concepts of this geometrical theory are outlined.

An analysis of science content and representations in introductory college physics textbooks and multimodal learning resources

NASA Astrophysics Data System (ADS)

Donnelly, Suzanne M.

This study features a comparative descriptive analysis of the physics content and representations surrounding the first law of thermodynamics as presented in four widely used introductory college physics textbooks representing each of four physics textbook categories (calculus-based, algebra/trigonometry-based, conceptual, and technical/applied). Introducing and employing a newly developed theoretical framework, multimodal generative learning theory (MGLT), an analysis of the multimodal characteristics of textbook and multimedia representations of physics principles was conducted. The modal affordances of textbook representations were identified, characterized, and compared across the four physics textbook categories in the context of their support of problem-solving. Keywords: college science, science textbooks, multimodal learning theory, thermodynamics, representations

Basic exploration geophysics

DOE Office of Scientific and Technical Information (OSTI.GOV)

Robinson, E.S.

1988-01-01

An introduction to geophysical methods used to explore for natural resources and to survey earth's geology is presented in this volume. It is suitable for second-and third-year undergraduate students majoring in geology or engineering and for professional engineering and for professional engineers and earth scientists without formal instruction in geophysics. The author assumes the reader is familiar with geometry, algebra, and trigonometry. Geophysical exploration includes seismic refraction and reflection surveying, electrical resistivity and electromagnetic field surveying, and geophysical well logging. Surveying operations are described in step-by-step procedures and are illustrated by practical examples. Computer-based methods of processing and interpreting datamore » as well as geographical methods are introduced.« less

The Effectiveness of Project Based Learning in Trigonometry

NASA Astrophysics Data System (ADS)

Gerhana, M. T. C.; Mardiyana, M.; Pramudya, I.

2017-09-01

This research aimed to explore the effectiveness of Project-Based Learning (PjBL) with scientific approach viewed from interpersonal intelligence toward students’ achievement learning in mathematics. This research employed quasi experimental research. The subjects of this research were grade X MIPA students in Sleman Yogyakarta. The result of the research showed that project-based learning model is more effective to generate students’ mathematics learning achievement that classical model with scientific approach. This is because in PjBL model students are more able to think actively and creatively. Students are faced with a pleasant atmosphere to solve a problem in everyday life. The use of project-based learning model is expected to be the choice of teachers to improve mathematics education.

New method: calculation of magnification factor from an intracardiac marker.

PubMed

Cha, S D; Incarvito, J; Maranhao, V

1983-01-01

In order to calculate a magnification factor (MF), an intracardiac marker (pigtail catheter with markers) was evaluated using a new formula and correlated with the conventional grid method. By applying the Pythagorean theorem and trigonometry, a new formula was developed, which is (formula; see text) In an experimental study, MF by the intracardiac markers was 0.71 +/- 0.15 (M +/- SD) and one by the grid method was 0.72 +/- 0.15, with a correlation coefficient of 0.96. In patients study, MF by the intracardiac markers was 0.77 +/- 0.06 and one by the grid method was 0.77 +/- 0.05. We conclude that this new method is simple and the results were comparable to the conventional grid method at mid-chest level.

Solving subsurface structural problems using a computer

DOE Office of Scientific and Technical Information (OSTI.GOV)

Witte, D.M.

1987-02-01

Until recently, the solution of subsurface structural problems has required a combination of graphical construction, trigonometry, time, and patience. Recent advances in software available for both mainframe and microcomputers now reduce the time and potential error of these calculations by an order of magnitude. Software for analysis of deviated wells, three point problems, apparent dip, apparent thickness, and the intersection of two planes, as well as the plotting and interpretation of these data can be used to allow timely and accurate exploration or operational decisions. The available computer software provides a set of utilities, or tools, rather than a comprehensive,more » intelligent system. The burden for selection of appropriate techniques, computation methods, and interpretations still lies with the explorationist user.« less

Monte Carlo simulation of portal dosimetry on a rectilinear voxel geometry: a variable gantry angle solution.

PubMed

Chin, P W; Spezi, E; Lewis, D G

2003-08-21

A software solution has been developed to carry out Monte Carlo simulations of portal dosimetry using the BEAMnrc/DOSXYZnrc code at oblique gantry angles. The solution is based on an integrated phantom, whereby the effect of incident beam obliquity was included using geometric transformations. Geometric transformations are accurate within +/- 1 mm and +/- 1 degrees with respect to exact values calculated using trigonometry. An application in portal image prediction of an inhomogeneous phantom demonstrated good agreement with measured data, where the root-mean-square of the difference was under 2% within the field. Thus, we achieved a dose model framework capable of handling arbitrary gantry angles, voxel-by-voxel phantom description and realistic particle transport throughout the geometry.

Social network analysis of a project-based introductory physics course

NASA Astrophysics Data System (ADS)

Oakley, Christopher

2016-03-01

Research suggests that students benefit from peer interaction and active engagement in the classroom. The quality, nature, effect of these interactions is currently being explored by Physics Education Researchers. Spelman College offers an introductory physics sequence that addresses content and research skills by engaging students in open-ended research projects, a form of Project-Based Learning. Students have been surveyed at regular intervals during the second semester of trigonometry-based course to determine the frequency of interactions in and out of class. These interactions can be with current or past students, tutors, and instructors. This line of inquiry focuses on metrics of Social Network analysis, such as centrality of participants as well as segmentation of groups. Further research will refine and highlight deeper questions regarding student performance in this pedagogy and course sequence.

Specular Reflection from Rough Surfaces Revisited

NASA Astrophysics Data System (ADS)

Yasuda, Kensei; Kim, Alvin; Cho, Hayley; Timofejev, Timofej; Walecki, Wojciech J.; Klep, James; Edelson, Amy S.; Walecki, Abigail S.; Walecki, Eve S.; Walecki, Peter S.

2016-10-01

In his beautiful paper, Hasan Fakhruddin reported observations of mirror-like reflections in the rough surface of a ground glass plate. Similar effects have been recently employed for metrology of the roughness of optical diffusers used in modern light emitting device illumination systems. We report the observations of specular reflection in nontransparent rough surfaces at oblique angles, where roughness was treated as a variable. We present a simple trigonometry-based model explaining the observed phenomenon, which we experimentally validated using aluminum surfaces that have controlled roughness. The reported demonstration requires no special equipment, other than cellphone cameras, dielectric or metal plate, and sandpaper, and serves as an introduction to wave optics. This activity can be used to get further insight into everyday applications of wave optics for students already familiar with wave optics fundamentals.

New methods in iris recognition.

PubMed

Daugman, John

2007-10-01

This paper presents the following four advances in iris recognition: 1) more disciplined methods for detecting and faithfully modeling the iris inner and outer boundaries with active contours, leading to more flexible embedded coordinate systems; 2) Fourier-based methods for solving problems in iris trigonometry and projective geometry, allowing off-axis gaze to be handled by detecting it and "rotating" the eye into orthographic perspective; 3) statistical inference methods for detecting and excluding eyelashes; and 4) exploration of score normalizations, depending on the amount of iris data that is available in images and the required scale of database search. Statistical results are presented based on 200 billion iris cross-comparisons that were generated from 632500 irises in the United Arab Emirates database to analyze the normalization issues raised in different regions of receiver operating characteristic curves.

Calculation of the change in corneal astigmatism following cataract extraction.

PubMed

Cravy, T V

1979-01-01

Obtaining a minimal amount of postoperative astigmatism following cataract surgery is becoming increasingly important. One aspect of the patient's surgery which should not be overlooked is the preoperative keratometry which provides a basis for preoperative planning of surgical technique to be used and a point of reference for determining the amount of change in astigmatism produced by the surgery. Analysis of the surgically induced change in astigmatism using the calculations described in this paper will allow the surgeon to evaluate his own techniques and to maximize his potential for obtaining consistently good postoperative astigmatic results without the need for suture removal. The method presented is based upon concepts in common use in surgical ophthalmology and requires only simple mathematical procedures, familiar to all with a background in algebra and trigonometry.

TG (Tri-Goniometry) technique: Obtaining perfect angles in Z-plasty planning with a simple ruler.

PubMed

Görgülü, Tahsin; Olgun, Abdulkerim

2016-03-01

The Z-plasty is used frequently in hand surgery to release post-burn scar contractures. Correct angles and equalization of each limb are the most important parts of the Z-plasty technique. A simple ruler is enough for equalization of limb but a goniometer is needed for accuracy and equalization of angles. Classically, angles of 30°, 45°, 60°, 75°, and 90° are used. These angles are important when elongating a contracture line or decreasing tension. Our method uses only trigonometry coefficients and a simple ruler, which is easily obtained and sterilized, enabling surgeons to perform all types of Z-plasty perfectly without measuring angles using a goniometer. Copyright © 2015 Elsevier Ltd and ISBI. All rights reserved.

Erosion Resistance Index (ERI) to Assess Surface Stability in Desert Environments

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hamada, Yuki; Grippo, Mark A.

2015-11-01

A new spectral index—erosion resistance index (ERI)—was developed to assess erosion risks in desert landscapes. The index was developed by applying trigonometry to the combination of the green/red band-ratio and the red/near infrared band-ratio from very high spatial resolution imagery. The resultant ERI maps showed spatially cohesive distributions of high and low index values across the study areas. High index values were observed over areas that were resistant to erosion (such as desert pavement and dense vegetation), while low index values overlapped with areas likely dominated by loose sandy soils, such as stream beds and access roads. Although further investigationmore » is warranted, this new index, ERI, shows promise for the assessment of erosion risks in desert regions.« less

STEREO/Waves Education and Public Outreach

NASA Astrophysics Data System (ADS)

MacDowall, R. J.; Bougeret, J.; Bale, S. D.; Goetz, K.; Kaiser, M. L.

2005-05-01

We present the education and public outreach plan and activities of the STEREO Waves (aka SWAVES) investigation. SWAVES measures radio emissions from the solar corona, interplanetary medium, and terrestrial magnetosphere, as well as in situ waves in the solar wind. In addition to the web site components that display stereo/multi-spacecraft data in a graphical form and explain the science and instruments, we will focus on the following three areas of EPO: class-room demonstrations using models of the STEREO spacecraft with battery powered radio receivers (and speakers) to illustrate spacecraft radio direction finding, teacher developed and tested class-room activities using SWAVES solar radio observations to motivate geometry and trigonometry, and sound-based delivery of characteristic radio and plasma wave events from the SWAVES web site for accessibility and esthetic reasons. Examples of each element will be demonstrated.

Introductio ad Geographiam et Sphaeram, A Lesser-Known Book on Astronomy

NASA Astrophysics Data System (ADS)

Stavinschi, Magda

2017-10-01

There are more than 300 years since Hrisant Nottara's book Introductio ad geographiam et sphaeram was first published in Paris. The book has already been translated into English by Greek colleagues in order to help contemporary readers get an insight into the scientific knowledge of that time and discover the scientific contributions of a priest who by the end of his career had become the Patriarch of Jerusalem. The authors aim to portray Nottara since he was the private tutor of the Wallachian prince C. Brancoveanu's sons, then working with Cassini I, until his final years when he became the leader of the church. A number of chapters of the book will be analysed in order to shed more light on the progress of science at the time, particularly in astronomy, trigonometry and geography

Exploring Magnetic Fields with a Compass

NASA Astrophysics Data System (ADS)

Lunk, Brandon; Beichner, Robert

2011-01-01

A compass is an excellent classroom tool for the exploration of magnetic fields. Any student can tell you that a compass is used to determine which direction is north, but when paired with some basic trigonometry, the compass can be used to actually measure the strength of the magnetic field due to a nearby magnet or current-carrying wire. In this paper, we present a series of simple activities adapted from the Matter & Interactions textbook for doing just this. Interestingly, these simple measurements are comparable to predictions made by the Bohr model of the atom. Although antiquated, Bohr's atom can lead the way to a deeper analysis of the atomic properties of magnets. Although originally developed for an introductory calculus-based course, these activities can easily be adapted for use in an algebra-based class or even at the high school level.

Le contenu astronomique des Sphériques de Ménélaos

NASA Astrophysics Data System (ADS)

Nadal, Robert; Taha, Abdelkaddous; Pinel, Pierre

2004-07-01

The Spherics were written by Menelaos in the form of a purely mathematical treatise. However, the material developed in the second and third book is closely linked to problems met in astronomy: computation of equatorial coordinates of the Sun, setting up of rising-time tables, study of the motion of the Sun in the sphaera obliqua, simultaneous risings. This link, which remains implicit in the text, was clearly displayed by two arabo-islamic mathematicians and astronomers, who expounded the astronomical meaning of some theorems of the Spherics. We describe, comment and complement their explanations, by classifying the implications of the theorems in three groups: direction of variation of some quantities on the sphere, spherical trigonometry and applications, direction of variation of ratios of some quantities on the sphere. An erratum to this article can be found at http://dx.doi.org/10.1007/s00407-004-0084-7

A model for precalculus students to determine the resonance frequency of a trumpet mouthpiece

NASA Astrophysics Data System (ADS)

Chapman, Robert C.

2004-05-01

The trumpet mouthpiece as a Helmholtz resonator is used to show precalculus students a mathematical model for determining the approximate resonance frequency of the mouthpiece. The mathematics is limited to algebra and trigonometry. Using a system of mouthpieces that have interchangeable cups and backbores, students are introduced to the acoustics of this resonator. By gathering data on 51 different configurations of mouthpieces, the author modifies the existing Helmholtz resonator equation to account for both cup volumes and backbore configurations. Students then use this model for frequency predictions. Included are how to measure the different physical attributes of a trumpet mouthpiece at minimal cost. This includes methods for measuring cup volume, backbore volume, backbore length, throat area, etc. A portion of this phase is de-signed for students to become acquainted with some of the vocabulary of acoustics and the physics of sound.

Morphology analysis of a foldable kirigami structure based on Miura origami

NASA Astrophysics Data System (ADS)

Jianguo, Cai; Xiaowei, Deng; Jian, Feng

2014-09-01

The morphology of a foldable kirigami structure with modified Miura-ori patterns, which displays curvature during motion, was investigated in this paper. The principle of spherical trigonometry was used to obtain the radius, span, rise, and longitudinal length of the foldable structure during motion. The results show that the radius of curvatures decreases and that the span initially increases and then decreases during the deployment process. Furthermore, there is little change in the span over the greater part of the deployment range. Changing the values for the length, a, and the vertex angle, β, demonstrates that the deployment angle at the end of the motion, the span, and the maximal rise increase with the increase in the length a. However, changing these values has no effect on the longitudinal length. At the same time, the effect of the vertex angle β on the geometry of the foldable kirigami is not significant.

The Challenge of Learning Physics Before Mathematics: A Case Study of Curriculum Change in Taiwan

NASA Astrophysics Data System (ADS)

Chiu, Mei-Shiu

2016-12-01

The aim of this study was to identify challenges in implementing a physics-before- 10 mathematics curriculum. Obviously, students need to learn necessary mathematics skills in order to develop advanced physics knowledge. In the 2010 high school curriculum in Taiwan, however, grade 11 science students study two-dimensional motion in physics without prior learning experiences of trigonometry in mathematics. The perspectives of three curriculum developers, 22 mathematics and physics teachers, two principals, and 45 science students were obtained by interview. The results of qualitative data analysis revealed six challenges and suggested likely solutions. The national level includes political and social challenges, resolved by respecting teachers as professionals; the teacher level includes knowledge and teaching challenges, resolved by increasing teacher trans-literal capacities; and the student level includes learning and justice challenges, resolved by focusing on students' diverse developments in cross-domain learning.

Castro Valley High School's Solar Panels

NASA Astrophysics Data System (ADS)

Lew, A.; Ham, S.; Shin, Y.; Yang, W.; Lam, J.

2014-12-01

Solar panels are photovoltaic cells that are designed to convert the sun's kinetic energy to generate usable energy in the form of electricity. Castro Valley High School has tried to offset the cost of electricity by installing solar panels, costing the district approximately 3.29 million dollars, but have been installed incorrectly and are not operating at peak efficency. By using trigonometry we deduced that Castro Valley High School's south facing solar panels were at an incline of 10o and that the east and west facing solar panels are at an incline of 5o. By taking the averages of the optimum angles for the months of September through May, roughly when school is in session, we found that the optimum angle for south facing solar panels should be roughly 46o. This shows that Castro Valley High School has not used it's budget to its full potential due to the fact that the solar panels were haphazardly installed.

Interrelationships Between 3 Keratoconic Cone Parameters.

PubMed

Tu, Kyaw L; Tourkmani, Abdo K; Srinivas, Singaram

2017-09-01

To find out the interrelationships between 3 parameters of the keratoconic cone. A total of 101 keratoconic eyes of 58 patients were included in this retrospective case series study. A complete eye examination was performed. Kmax (K) and pachymetry at the thinnest point (T) were obtained from the Pentacam tomographer. The vertex to thinnest pachymetry distance (D for decentration) was calculated using trigonometry. Pearson correlation coefficients between T and D, between T and K, and between D and K were calculated. There is a statistically significant positive correlation between thinnest point pachymetry and decentration (R = 0.366, P = 0.0002) and also statistically significant negative correlation between thinnest point pachymetry and Kmax (R = -0.719, P < 0.00001) and decentration and Kmax (R = -0.281, P = 0.0044). The interrelationships between the 3 keratoconic cone parameters suggest that the thinner cones are largely central, that is, decenter less, but show greater steepening.

Recommended coordinate systems for thin spherocylindrical lenses.

PubMed

Deal, F C; Toop, J

1993-05-01

Because the set of thin spherocylindrical lenses forms a vector space, any such lens can be expressed in terms of its cartesian coordinates with respect to whatever set of basis lenses we may choose. Two types of cartesian coordinate systems have become prominent, those having coordinates associated with the lens power matrix and those having coordinates associated with the Humphrey Vision Analyzer. This paper emphasizes the value of a particular cartesian coordinate system of the latter type, and the cylindrical coordinate system related to it, by showing how it can simplify the trigonometry of adding lenses and how it preserves symmetry in depicting the sets of all spherical lenses, all Jackson crossed-cylinders, and all cylindrical lenses. It also discusses appropriate coordinates for keeping statistics on lenses and shows that an easy extension of the lens vector space to include general optical systems is not possible.

Prediction by computerised tomography of distance from skin to epidural space during thoracic epidural insertion.

PubMed

Carnie, J; Boden, J; Gao Smith, F

2002-07-01

In this single group observational study on 29 patients, we describe a technique that predicts the depth of the epidural space, calculated from the routine pre-operative chest computerised tomography (CT) scan using Pythagorean triangle trigonometry. We also compared the CT-derived depth of the epidural space with the actual depth of needle insertion. The CT-derived and the actual depths of the epidural space were highly correlated (r = 0.88, R2 = 0.78, p < 0.0001). The mean (95% CI) difference between CT-derived and actual depths was 0.26 (0.03-0.49) cm. Thus, the CT-derived depth tends to be greater than the actual depth by between 0.03 and 0.49 cm. There were no associations between either the CT-derived or the actual depth of the epidural space and age, weight, height or body mass index.

The Impact of the Louisiana State University Physics Entrance Requirement on Secondary Physics in Louisiana

NASA Astrophysics Data System (ADS)

McCoy, Michael Hanson

State Department of Education data was examined to determine the number of students enrolled in physics, physics class number, physics teacher number, and physics teacher certification. Census data from public and nonpublic school teachers, principals, and superintendents was analyzed. Purposive sampling of seven public and four nonpublic schools was used for site visitation including observations of physics classes, interviews of teachers and principals, and document acquisition. The literature base was drawn from a call for an increase in academic requirements in the sciences by the National Commission on Excellence in Education, the Southern Regional Education Board, the American Association for Advancement in the Sciences, and numerous state boards of education. LSU is the only major state university to require physics as an academic admission standard. Curriculum changes which influenced general curriculum change were: leveling of physics classes; stressing concepts, algebra, and doing problems in level-one; stressing trigonometry and problem solving in level-two; and increased awareness of expectations for university admission. Certified physics teachers were positive toward the requirement. The majority adopted a "wait-and-see" attitude to see if the university would institute the physics standard. Some physics teachers, nonphysics majors, were opposed to the requirement. Those who were positive remained positive. Those who developed the wait-and-see adopted the leveled physics course concept in 1989 and were positive toward the requirement. College-bound physics was taught prior to the requirement. The State Department of Education leveled physics in 1989. Level-one physics was algebra and conceptual based, level-two physics was trigonometry based, and a level-three physics, advanced placement was added. Enrollment doubled in public schools and increased 40% in nonpublic schools. African-American enrollment almost doubled in public and nonpublic schools. Oriental enrollment increased 40% in public schools. Hispanic enrollment increased 120% in public schools. Female enrollment in public schools increased 27.6% and 10% in nonpublic schools. The number of physics faculty members increased 33% in public schools and 25% in nonpublic schools. Newly certified physics teachers increased 80% although demand exceeded teacher supply. The proportion of certified to noncertified public school physics teachers declined 12% and spiraled downward 25% for nonpublic school physics teachers.

Qibla Finder and Sholat Times Based on Digital Compass, GPS and Microprocessor

NASA Astrophysics Data System (ADS)

Sanjaya, W. S. M.; Anggraeni, D.; Nurrahman, F. I.; Kresnadjaja, W. G.; Dewi, I. P.; Mira; Aliah, H.; Marlina, L.

2018-01-01

To performing Sholat, Muslims around the world are required to pay attention to the requirements of Sholat, such as; determining the direction of the Qibla (Kaaba) and the time of Sholat. In this research will be made a real time Qibla Finder and Sholat Times named Q-Bot Ver3 to help Muslims find a Qibla direction and Time of Sholat anywhere. This Qibla Finder and Sholat Times are developed with robotic technology based on Digital Compass, GPS and Microcontroller. To determine the Qibla direction and Sholat times, latitude and longitude data form GPS module processed used spherical triangle trigonometry method, while the compass module used to show the Qibla direction. Moreover, this system has a buzzer which can sound if the device facing to the Qibla. This system is reliable and accurate in determining the Qibla Finder and Sholat Times. Thus, the advantage of the system is can correct the Qibla of Masjid and can help blind people to facing Qibla around the world.

Introducing the Moon's Orbital Eccentricity

NASA Astrophysics Data System (ADS)

Oostra, Benjamin

2014-11-01

I present a novel way to introduce the lunar orbital eccentricity in introductory astronomy courses. The Moon is perhaps the clearest illustration of the general orbital elements such as inclination, ascending node, eccentricity, perigee, and so on. Furthermore, I like the students to discover astronomical phenomena for themselves, by means of a guided exercise, rather than just telling them the facts.1 The inclination and nodes may be found by direct observation, monitoring carefully the position of the Moon among the stars. Even the regression of the nodes may be discovered in this way2 To find the eccentricity from students' observations is also possible,3 but that requires considerable time and effort. if a whole class should discover it in a short time, here is a method more suitable for a one-day class or home assignment. The level I aim at is, more or less, advanced high school or first-year college students. I assume them to be acquainted with celestial coordinates and the lunar phases, and to be able to use algebra and trigonometry.

DECOMPOSITION OF THE PARTICLE AND CONNECTION OF PARTICLES IN THE TERMINI OF THE MOMENTUM SPACE

DOE Office of Scientific and Technical Information (OSTI.GOV)

Chernikov, N.A.

1958-01-01

>Geometric and algebraic notions and ideas are used to obtain a geometric interpretation of the kinematics of nuclear reactions. Thus, extended analytic calculations combined with the transition from one reference system to another, are replaced by simple formulas of the hyperbolic trigonometry. Let a particle move with the velocity a in a reference system which moves with the velocity o. Then the modulus of the three-dimensional impulse of the particle is p/sub 0a/ = m c sh oa-bar/c, where m is the resting miss, c is the velocity of the light, oa-bar is the distance of the points o andmore » a in the momentum space. The kinetic energy epsilon /sub oa/ of the particle in the system o is epsilon / sub oa/=m c/sup 2/STAoa-bar/c-1!. Then the ratio epsilon /sub oa/m is the area divided by 2 pi of a circle of radius oa in the momentum space. (TCO)« less

Proposal for a trigonometric method to evaluate the abduction angle of the lower limbs in neonates.

PubMed

Lima, Thaís; Alves, Cyntia; Funayama, Carolina A R

2008-12-01

It is difficult to precisely measure articular arc movement in newborns using a goniometer. This article proposes an objective method based on trigonometry for the evaluation of lower limb abduction. With the newborn aligned in the dorsal decubitus position, 2 points are marked at the level of the medial malleolus, one on the sagittal line and the other at the end of the abduction. Using the right-sided line between these 2 points and a line from the medial malleolus to the reference point at the anterior superior iliac spine or umbilical scar, an isosceles triangle is drawn, and half of the inferential abduction angle is obtained by calculating the sine. Twenty healthy full-term newborns comprise the study cohort. Intersubject and intrasubject variability among the abduction angle values (mean [SD], 37 degrees [4] degrees) is low. This method is advantageous because the measurement is precise and because the sine can be used without approximation.

A Novel AMARS Technique for Baseline Wander Removal Applied to Photoplethysmogram.

PubMed

Timimi, Ammar A K; Ali, M A Mohd; Chellappan, K

2017-06-01

A new digital filter, AMARS (aligning minima of alternating random signal) has been derived using trigonometry to regulate signal pulsations inline. The pulses are randomly presented in continuous signals comprising frequency band lower than the signal's mean rate. Frequency selective filters are conventionally employed to reject frequencies undesired by specific applications. However, these conventional filters only reduce the effects of the rejected range producing a signal superimposed by some baseline wander (BW). In this work, filters of different ranges and techniques were independently configured to preprocess a photoplethysmogram, an optical biosignal of blood volume dynamics, producing wave shapes with several BWs. The AMARS application effectively removed the encountered BWs to assemble similarly aligned trends. The removal implementation was found repeatable in both ear and finger photoplethysmograms, emphasizing the importance of BW removal in biosignal processing in retaining its structural, functional and physiological properties. We also believe that AMARS may be relevant to other biological and continuous signals modulated by similar types of baseline volatility.

On quanta, mind, and matter: Hans Primas in Context. H. Atmanspacher, A. Amann, U. Müller-Herold (Eds), Kluwer, Boston, 1999, pp. 398 + viii, US192.00 £133.56 (hardback), ISBN 0-7923-5696-9

NASA Astrophysics Data System (ADS)

Halvorson, Hans

This book consists of a collection of 18 articles by students, colleagues, and friends of the theoretical chemist/philosopher of science, Hans Primas. Although the articles are written in honor of Primas, the editors state that the collection is not a Festschrift, but that it simply 'takes up some of the discussions that Primas has initiated or inspired' (p. 2). Due to the breadth of Primas' intellectual interests, the result is a highly eclectic collection of articles, whose topics range through (i) the historical development of experimental NMR spectroscopy, (ii) Jungian psychology, (iii) a defense of determinism in the rationalistic tradition of Leibniz, to (iv) a mathematical exposition of 'operator trigonometry.' However, since our space-and my expertise-is limited, the following comments will be restricted to those articles that deal with the mathematical and conceptual foundations of physics, a subject to which Primas has made a large contribution.

Apparatus and method for high-speed characterization of surfaces

NASA Technical Reports Server (NTRS)

Simmons, Stephen M. (Inventor); Hooker, Jeffery A. (Inventor)

1997-01-01

This invention permits rapid determination of a three-dimensional contour of an object. The apparatus directs a laser beam onto a point on an object, and the scattered light is received by a position sensitive detector (PSD). The location at which scattered light is received can be related to the displacement of the point on the object, using basic trigonometry. The laser beam is deflected non-mechanically, by passing it through a crystal whose index of refraction is varied in a controlled manner. By proper control of the index of refraction, one can scan the beam across a desired area of the object, without using any moving parts, and can therefore obtain complete information about the contour of the object. The device operates without making any conventional images, and is therefore much faster than comparable devices of the prior art. Also, the laser and beam deflector can be housed in a small optical head which enables the device to reach normally inaccessible areas. The device can compare the test object with a reference object, or it can compare two objects for differences.

``Simplest Molecule'' Clarifies Modern Physics II. Relativistic Quantum Mechanics

NASA Astrophysics Data System (ADS)

Harter, William; Reimer, Tyle

2015-05-01

A ``simplest molecule'' consisting of CW- laser beam pairs helps to clarify relativity from poster board - I. In spite of a seemingly massless evanescence, an optical pair also clarifies classical and quantum mechanics of relativistic matter and antimatter. Logical extension of (x,ct) and (ω,ck) geometry gives relativistic action functions of Hamiltonian, Lagrangian, and Poincare that may be constructed in a few ruler-and-compass steps to relate relativistic parameters for group or phase velocity, momentum, energy, rapidity, stellar aberration, Doppler shifts, and DeBroglie wavelength. This exposes hyperbolic and circular trigonometry as two sides of one coin connected by Legendre contact transforms. One is Hamiltonian-like with a longitudinal rapidity parameter ρ (log of Doppler shift). The other is Lagrange-like with a transverse angle parameter σ (stellar aberration). Optical geometry gives recoil in absorption, emission, and resonant Raman-Compton acceleration and distinguishes Einstein rest mass, Galilean momentum mass, and Newtonian effective mass. (Molecular photons appear less bullet-like and more rocket-like.) In conclusion, modern space-time physics appears as a simple result of the more self-evident Evenson's axiom: ``All colors go c.''

"simplest Molecule" Clarifies Modern Physics II. Relativistic Quantum Mechanics

NASA Astrophysics Data System (ADS)

Reimer, T. C.; Harter, W. G.

2014-06-01

A "simplest molecule" consisting of CW-laser beam pairs helps to clarify relativity in Talk I. In spite of a seemingly massless evanescence, an optical pair also clarifies classical and quantum mechanics of relativistic matter and anti-matter. *Logical extension of (x,ct) and (ω,ck) geometry gives relativistic action functions of Hamiltonian, Lagrangian, and Poincare that may be constructed in a few ruler-and-compass steps to relate relativistic parameters for group or phase velocity, momentum, energy, rapidity, stellar aberration, Doppler shifts, and DeBroglie wavelength. This exposes hyperbolic and circular trigonometry as two sides of one coin connected by Legendre contact transforms. One is Hamiltonian-like with a longitudinal rapidity parameter ρ (log of Doppler shift). The other is Lagrange-like with a transverse angle parameter σ (stellar aberration). Optical geometry gives recoil in absorption, emission, and resonant Raman-Compton acceleration and distinguishes Einstein rest mass, Galilean momentum mass, and Newtonian effective mass. (Molecular photons appear less bullet-like and more rocket-like.) In conclusion, modern space-time physics appears as a simple result of the more self-evident Evenson's axiom: "All colors go c."

A Study of Visualization for Mathematics Education

NASA Technical Reports Server (NTRS)

Daugherty, Sarah C.

2008-01-01

Graphical representations such as figures, illustrations, and diagrams play a critical role in mathematics and they are equally important in mathematics education. However, graphical representations in mathematics textbooks are static, Le. they are used to illustrate only a specific example or a limited set. of examples. By using computer software to visualize mathematical principles, virtually there is no limit to the number of specific cases and examples that can be demonstrated. However, we have not seen widespread adoption of visualization software in mathematics education. There are currently a number of software packages that provide visualization of mathematics for research and also software packages specifically developed for mathematics education. We conducted a survey of mathematics visualization software packages, summarized their features and user bases, and analyzed their limitations. In this survey, we focused on evaluating the software packages for their use with mathematical subjects adopted by institutions of secondary education in the United States (middle schools and high schools), including algebra, geometry, trigonometry, and calculus. We found that cost, complexity, and lack of flexibility are the major factors that hinder the widespread use of mathematics visualization software in education.

A power function profile of a ski jumping in-run hill.

PubMed

Zanevskyy, Ihor

2011-01-01

The aim of the research was to find a function of the curvilinear segment profile which could make possible to avoid an instantaneous increasing of a curvature and to replace a circle arc segment on the in-run of a ski jump without any correction of the angles of inclination and the length of the straight-line segments. The methods of analytical geometry and trigonometry were used to calculate an optimal in-run hill profile. There were two fundamental conditions of the model: smooth borders between a curvilinear segment and straight-line segments of an in-run hill and concave of the curvilinear segment. Within the framework of this model, the problem has been solved with a reasonable precision. Four functions of a curvilinear segment profile of the in-run hill were investigated: circle arc, inclined quadratic parabola, inclined cubic parabola, and power function. The application of a power function to the in-run profile satisfies equal conditions for replacing a circle arc segment. Geometrical parameters of 38 modern ski jumps were investigated using the methods proposed.

The usage of image trigonometry in bone measurements.

PubMed

Dymond, Ian W; Ashforth, James A; Dymond, Graeme F; Spirakis, Thanos; Learmonth, Ian D

2013-01-01

The entire musculo-skeletal system responds dynamically to stresses and strains applied to it. Restoring normal biomechanics contributes to the normal function that ensures that physiological stresses and strains are preserved. Appropriate preoperative planning is mandatory to restore normal biomechanics at reconstructive surgery. Effective preoperative planning depends on the ability to reproducibly make accurate measurements of lengths and angles from plain radiographs. Measurement has become an integral part of orthopaedics to define morphological abnormality, to plan for reconstruction and for comparative research. The most prevalent method of measurement is usually based on lines drawn on radiographs with no accurate reference to the actual geometry of the structures. This two-dimensional projection of an asymmetrical three-dimensional structure leads to inaccuracy and consequently to a compromise in the overall precision of many procedures. In addition it is also difficult to monitor the progression of disease as the exact relationship of the bones and joints to each other, and to prosthetics, cannot be accurately recorded. This paper presents a method of digitally measuring relevant bone parameters in a geometric manner in order to achieve accurate, repeatable measurements.

Investigating adaptive reasoning and strategic competence: Difference male and female

NASA Astrophysics Data System (ADS)

Syukriani, Andi; Juniati, Dwi; Siswono, Tatag Yuli Eko

2017-08-01

The series of adaptive reasoning and strategic competencies represent the five components of mathematical proficiency to describe the students' mathematics learning success. Gender contribute to the problem-solving process. This qualitative research approach investigated the adaptive reasoning and strategic competence aspects of a male student and a female student when they solved mathematical problem. They were in the eleventh grade of high school in Makassar. Both also had similar mathematics ability and were in the highest category. The researcher as the main instrument used secondary instrument to obtain the appropriate subject and to investigate the aspects of adaptive reasoning and strategic competence. Test of mathematical ability was used to locate the subjects with similar mathematical ability. The unstructured guideline interview was used to investigate aspects of adaptive reasoning and strategic competence when the subject completed the task of mathematical problem. The task of mathematical problem involves several concepts as the right solution, such as the circle concept, triangle concept, trigonometry concept, and Pythagoras concept. The results showed that male and female subjects differed in applying a strategy to understand, formulate and represent the problem situation. Furthermore, both also differed in explaining the strategy used and the relationship between concepts and problem situations.

Love and fear of heights: the pathophysiology and psychology of height imbalance.

PubMed

Salassa, John R; Zapala, David A

2009-01-01

Individual psychological responses to heights vary on a continuum from acrophobia to height intolerance, height tolerance, and height enjoyment. This paper reviews the English literature and summarizes the physiologic and psychological factors that generate different responses to heights while standing still in a static or motionless environment. Perceptual cues to height arise from vision. Normal postural sway of 2 cm for peripheral objects within 3 m increases as eye-object distance increases. Postural sway >10 cm can result in a fall. A minimum of 20 minutes of peripheral retinal arc is required to detect motion. Trigonometry dictates that a 20-minute peripheral retinal arch can no longer be achieved in a standing position at an eye-object distance of >20 m. At this distance, visual cues conflict with somatosensory and vestibular inputs, resulting in variable degrees of imbalance. Co-occurring deficits in the visual, vestibular, and somatosensory systems can significantly increase height imbalance. An individual's psychological makeup, influenced by learned and genetic factors, can influence reactions to height imbalance. Enhancing peripheral vision and vestibular, proprioceptive, and haptic functions may improve height imbalance. Psychotherapy may improve the troubling subjective sensations to heights.

The Infinite in the Finite

NASA Astrophysics Data System (ADS)

Macintosh Wilson, Alistair

1996-01-01

A conversation between Euclid and the ghost of Socrates. . . the paths of the moon and the sun charted by the stone-builders of ancient Europe. . .the Greek ideal of the golden mean by which they measured beauty. . . Combining historical fact with a retelling of ancient myths and legends, this lively and engaging book describes the historical, religious and geographical background that gave rise to mathematics in ancient Egypt, Babylon, China, Greece, India, and the Arab world. Each chapter contains a case study where mathematics is applied to the problems of the era, including the area of triangles and volume of the Egyptian pyramids; the Babylonian sexagesimal number system and our present measure of space and time which grew out of it; the use of the abacus and remainder theory in China; the invention of trigonometry by Arab mathematicians; and the solution of quadratic equations by completing the square developed in India. These insightful commentaries will give mathematicians and general historians a better understanding of why and how mathematics arose from the problems of everyday life, while the author's easy, accessible writing style will open fascinating chapters in the history of mathematics to a wide audience of general readers.

Precalculus teachers' perspectives on using graphing calculators: an example from one curriculum

NASA Astrophysics Data System (ADS)

Karadeniz, Ilyas; Thompson, Denisse R.

2018-01-01

Graphing calculators are hand-held technological tools currently used in mathematics classrooms. Teachers' perspectives on using graphing calculators are important in terms of exploring what teachers think about using such technology in advanced mathematics courses, particularly precalculus courses. A descriptive intrinsic case study was conducted to analyse the perspectives of 11 teachers using graphing calculators with potential Computer Algebra System (CAS) capability while teaching Functions, Statistics, and Trigonometry, a precalculus course for 11th-grade students developed by the University of Chicago School Mathematics Project. Data were collected from multiple sources as part of a curriculum evaluation study conducted during the 2007-2008 school year. Although all teachers were using the same curriculum that integrated CAS into the instructional materials, teachers had mixed views about the technology. Graphing calculator features were used much more than CAS features, with many teachers concerned about the use of CAS because of pressures from external assessments. In addition, several teachers found it overwhelming to learn a new technology at the same time they were learning a new curriculum. The results have implications for curriculum developers and others working with teachers to update curriculum and the use of advanced technologies simultaneously.

Periodicity of microfilariae of human filariasis analysed by a trigonometric method (Aikat and Das).

PubMed

Tanaka, H

1981-04-01

The microfilarial periodicity of human filariae was characterized statistically by fitting the observed change of microfilaria (mf) counts to the formula of a simple harmonic wave using two parameters, the peak hour (K) and periodicity index (D) (Sasa & Tanaka, 1972, 1974). Later Aikat and Das (1976) proposed a simple calculation method using trigonometry (A-D method) to determine the peak hour (K) and periodicity index (P). All data of microfilarial periodicity analysed previously by the method of Sasa and Tanaka (S-T method) were calculated again by the A-D method in the present study to evaluate the latter method. The results of calculations showed that P was not proportional to D and the ratios of P/D were mostly smaller than expected, especially when P or D was small in less periodic forms. The peak hour calculated by the A-D method did not differ much from that calculated by the S-T method. Goodness of fit was improved slightly by the A-K method in two thirds of analysed data. The classification of human filariae in respect of the type of periodicity was, however, changed little by the results calculated by the A-D method.

Program helps quickly calculate deviated well path

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gardner, M.P.

1993-11-22

A BASIC computer program quickly calculates the angle and measured depth of a simple directional well given only the true vertical depth and total displacement of the target. Many petroleum engineers and geologists need a quick, easy method to calculate the angle and measured depth necessary to reach a target in a proposed deviated well bore. Too many of the existing programs are large and require much input data. The drilling literature is full of equations and methods to calculate the course of well paths from surveys taken after a well is drilled. Very little information, however, covers how tomore » calculate well bore trajectories for proposed wells from limited data. Furthermore, many of the equations are quite complex and difficult to use. A figure lists a computer program with the equations to calculate the well bore trajectory necessary to reach a given displacement and true vertical depth (TVD) for a simple build plant. It can be run on an IBM compatible computer with MS-DOS version 5 or higher, QBasic, or any BASIC that does no require line numbers. QBasic 4.5 compiler will also run the program. The equations are based on conventional geometry and trigonometry.« less

[Anatomic foundation of the lateral portal for radiotherapy of nasopharyngeal cancer (NPC)].

PubMed

Wei, B Q; Feng, P B; Li, J Z

1987-05-01

Basing on 31 normal skulls, the lateral projections of some points relative to the bony structure near the nasopharynx were located under the simulator, followed by drawing it on a sheet of paper with the aid of geometry and trigonometry. Thus, the relation between external and internal structures is shown on the drawn projection, which can serve as the anatomic basis for designing the routine field and improving radiotherapy technique. In the light of data informed by this study and clinical experiences of the authors and others, it was found logical, in radiotherapy of NPC, that large opposing lateral pre-auriculo-cervical portals with their posterior margin extending beyond the external auditory meatus posteriorly be used in order to avoid geographic miss of the uppermost deep cervical lymph nodes usually involved beneath the jugular foramen and posterior portion of the nasopharynx. In addition, the upper margin of the lateral portal must be parallel but superior to the cantho-auditory line, on which the foramen ovale is projected. Actual locating the upper margin should depend on the extent of the intracranial invasion of the tumor as shown by the CT scan.

Optimizing the positional relationships between instruments used in laparoscopic simulation using a simple trigonometric method.

PubMed

Lorias Espinoza, Daniel; Ordorica Flores, Ricardo; Minor Martínez, Arturo; Gutiérrez Gnecchi, José Antonio

2014-06-01

Various methods for evaluating laparoscopic skill have been reported, but without detailed information on the configuration used they are difficult to reproduce. Here we present a method based on the trigonometric relationships between the instruments used in a laparoscopic training platform in order to provide a tool to aid in the reproducible assessment of surgical laparoscopic technique. The positions of the instruments were represented using triangles. Basic trigonometry was used to objectively establish the distances among the working ports RL, the placement of the optical port h', and the placement of the surgical target OT. The optimal configuration of a training platform depends on the selected working angles, the intracorporeal/extracorporeal lengths of the instrument, and the depth of the surgical target. We demonstrate that some distances, angles, and positions of the instruments are inappropriate for satisfactory laparoscopy. By applying basic trigonometric principles we can determine the ideal placement of the working ports and the optics in a simple, precise, and objective way. In addition, because the method is based on parameters known to be important in both the performance and quantitative quality of laparoscopy, the results are generalizable to different training platforms and types of laparoscopic surgery.

Grid cell hexagonal patterns formed by fast self-organized learning within entorhinal cortex.

PubMed

Mhatre, Himanshu; Gorchetchnikov, Anatoli; Grossberg, Stephen

2012-02-01

Grid cells in the dorsal segment of the medial entorhinal cortex (dMEC) show remarkable hexagonal activity patterns, at multiple spatial scales, during spatial navigation. It has previously been shown how a self-organizing map can convert firing patterns across entorhinal grid cells into hippocampal place cells that are capable of representing much larger spatial scales. Can grid cell firing fields also arise during navigation through learning within a self-organizing map? This article describes a simple and general mathematical property of the trigonometry of spatial navigation which favors hexagonal patterns. The article also develops a neural model that can learn to exploit this trigonometric relationship. This GRIDSmap self-organizing map model converts path integration signals into hexagonal grid cell patterns of multiple scales. GRIDSmap creates only grid cell firing patterns with the observed hexagonal structure, predicts how these hexagonal patterns can be learned from experience, and can process biologically plausible neural input and output signals during navigation. These results support an emerging unified computational framework based on a hierarchy of self-organizing maps for explaining how entorhinal-hippocampal interactions support spatial navigation. Copyright © 2010 Wiley Periodicals, Inc.

Route choice in mountain navigation, Naismith's rule, and the equivalence of distance and climb.

PubMed

Scarf, Philip

2007-04-01

In this paper, I consider decision making about routes in mountain navigation. In particular, I discuss Naismith's rule, a method of calculating journey times in mountainous terrain, and its use for route choice. The rule is essentially concerned with the equivalence, in terms of time duration, between climb or ascent and distance travelled. Naismith himself described a rule that is purported to be based on trigonometry and simple assumptions about rate of ascent; his rule with regard to hill-walking implies that 1 m of ascent is equivalent to 7.92 m of horizontal travel (1:7.92). The analysis of data on fell running records presented here supports Naismith's rule and it is recommended that male runners and walkers use a 1:8 equivalence ratio and females a 1:10 ratio. The present findings are contrasted with those based on the analysis of data relating to treadmill running experiments (1:3.3), and with those based on the analysis of times for a mountain road-relay (1:4.4). Analysis of cycling data suggests a similar rule (1:8.2) for cycling on mountainous roads and tracks.

Experimental determination of torsion angles in the polypeptide backbone of the gramicidin A channel by solid state nuclear magnetic resonance.

PubMed

Teng, Q; Nicholson, L K; Cross, T A

1991-04-05

An analytical method for the determination of torsion angles from solid state 15N nuclear magnetic resonance (n.m.r.) spectroscopic data is demonstrated. Advantage is taken of the 15N-1H and 15N-13C dipolar interactions as well as the 15N chemical shift interaction in oriented samples. The membrane-bound channel conformation of gramicidin A has eluded an atomic resolution structure determination by more traditional approaches. Here, the torsion angles for the Ala3 site are determined by obtaining the n.m.r. data for both the Gly2-Ala3 and Ala3-Leu4 peptide linkages. Complete utilization of the orientational constraints derived from these orientation-dependent nuclear spin interactions in restricting the conformational space is most effectively achieved by utilizing spherical trigonometry. Two possible sets of torsion angles for the Ala3 site are obtained (phi, psi = -129 degrees, 153 degrees and -129 degrees, 122 degrees), both of which are consistent with a right-handed beta-helix. Other functional and computational evidence strongly supports the set for which the carbonyl oxygen atom of the Ala3-Leu4 linkage is rotated into the channel lumen.

Precise montaging and metric quantification of retinal surface area from ultra-widefield fundus photography and fluorescein angiography.

PubMed

Croft, Daniel E; van Hemert, Jano; Wykoff, Charles C; Clifton, David; Verhoek, Michael; Fleming, Alan; Brown, David M

2014-01-01

Accurate quantification of retinal surface area from ultra-widefield (UWF) images is challenging due to warping produced when the retina is projected onto a two-dimensional plane for analysis. By accounting for this, the authors sought to precisely montage and accurately quantify retinal surface area in square millimeters. Montages were created using Optos 200Tx (Optos, Dunfermline, U.K.) images taken at different gaze angles. A transformation projected the images to their correct location on a three-dimensional model. Area was quantified with spherical trigonometry. Warping, precision, and accuracy were assessed. Uncorrected, posterior pixels represented up to 79% greater surface area than peripheral pixels. Assessing precision, a standard region was quantified across 10 montages of the same eye (RSD: 0.7%; mean: 408.97 mm(2); range: 405.34-413.87 mm(2)). Assessing accuracy, 50 patients' disc areas were quantified (mean: 2.21 mm(2); SE: 0.06 mm(2)), and the results fell within the normative range. By accounting for warping inherent in UWF images, precise montaging and accurate quantification of retinal surface area in square millimeters were achieved. Copyright 2014, SLACK Incorporated.

Correlation between polar values and vector analysis.

PubMed

Naeser, K; Behrens, J K

1997-01-01

To evaluate the possible correlation between polar value and vector analysis assessment of surgically induced astigmatism. Department of Ophthalmology, Aalborg Sygehus Syd, Denmark. The correlation between polar values and vector analysis was evaluated by simple mathematical and optical methods using accepted principles of trigonometry and first-order optics. Vector analysis and polar values report different aspects of surgically induced astigmatism. Vector analysis describes the total astigmatic change, characterized by both astigmatic magnitude and direction, while the polar value method produces a single, reduced figure that reports flattening or steepening in preselected directions, usually the plane of the surgical meridian. There is a simple Pythagorean correlation between vector analysis and two polar values separated by an arch of 45 degrees. The polar value calculated in the surgical meridian indicates the power or the efficacy of the surgical procedure. The polar value calculated in a plane inclined 45 degrees to the surgical meridian indicates the degree of cylinder rotation induced by surgery. These two polar values can be used to obtain other relevant data such as magnitude, direction, and sphere of an induced cylinder. Consistent use of these methods will enable surgeons to control and in many cases reduce preoperative astigmatism.

Exactly Solvable Models in Many-Body Theory

NASA Astrophysics Data System (ADS)

March, N. H.; Angilella, G. G. N.

2016-06-01

This book is an introduction to wave dynamics as they apply to earthquakes, among the scariest, most unpredictable, and deadliest natural phenomena on Earth. Since studying seismic activity is essentially a study of wave dynamics, this text starts with a discussion of types and representations, including wave-generation mechanics, superposition, and spectral analysis. Simple harmonic motion is used to analyze the mechanisms of wave propagation, and driven and damped systems are used to model the decay rates of various modal frequencies in different media. Direct correlation to earthquakes in California, Mexico, and Japan is used to illustrate key issues, and actual data from an event in California is presented and analyzed. Our Earth is a dynamic and changing planet, and seismic activity is the result. Hundreds of waves at different frequencies, modes, and amplitudes travel through a variety of different media, from solid rock to molten metals. Each media responds differently to each mode; consequently the result is an enormously complicated dynamic behavior. Earthquakes should serve well as a complimentary text for an upper-school course covering waves and wave mechanics, including sound and acoustics and basic geology. The mathematical requirement includes trigonometry and series summations, which should be accessible to most upper-school and college students. Animation, sound files, and videos help illustrate major topics.

Use of the Brainlab Disposable Stylet for endoscope and peel-away navigation.

PubMed

Halliday, Jane; Kamaly, Ian

2016-12-01

Neuronavigation, the ability to perform real-time intra-operative guidance during cranial and/or spinal surgery, has increased both accuracy and safety in neurosurgery [2]. Cranial navigation of existing surgical instruments using Brainlab requires the use of an instrument adapter and clamp, which in our experience renders an endoscope 'top-heavy', difficult to manipulate, and the process of registration of the adapter quite time-consuming. A Brainlab Disposable Stylet was used to navigate fenestration of an entrapped temporal horn in a pediatric case. Accuracy was determined by target visualization relative to neuronavigation targeting. Accuracy was also calculated using basic trigonometry to establish the maximum tool tip inaccuracy for the disposible stylet inserted into a peel-away (Codman) and endoscope. The Brainlab Disposable Stylet was easier to use, more versatile, and as accurate as use of an instrument adapter and clamp. The maximum tool-tip inaccuracy for the endoscope was 0.967 mm, and the Codman peel-away 0.489 mm. A literature review did not reveal any reports of use of the Brainlab Disposable Stylet in this way, and we are unaware of this being used in common neurosurgical practice. We would recommend this technique in endoscopic cases that require use of Brainlab navigation.

Concept mapping learning strategy to enhance students' mathematical connection ability

NASA Astrophysics Data System (ADS)

Hafiz, M.; Kadir, Fatra, Maifalinda

2017-05-01

The concept mapping learning strategy in teaching and learning mathematics has been investigated by numerous researchers. However, there are still less researchers who have scrutinized about the roles of map concept which is connected to the mathematical connection ability. Being well understood on map concept, it may help students to have ability to correlate one concept to other concept in order that the student can solve mathematical problems faced. The objective of this research was to describe the student's mathematical connection ability and to analyze the effect of using concept mapping learning strategy to the students' mathematical connection ability. This research was conducted at senior high school in Jakarta. The method used a quasi-experimental with randomized control group design with the total number was 72 students as the sample. Data obtained through using test in the post-test after giving the treatment. The results of the research are: 1) Students' mathematical connection ability has reached the good enough level category; 2) Students' mathematical connection ability who had taught with concept mapping learning strategy is higher than who had taught with conventional learning strategy. Based on the results above, it can be concluded that concept mapping learning strategycould enhance the students' mathematical connection ability, especially in trigonometry.

A Worksheet to Enhance Students’ Conceptual Understanding in Vector Components

NASA Astrophysics Data System (ADS)

Wutchana, Umporn; Emarat, Narumon

2017-09-01

With and without physical context, we explored 59 undergraduate students’conceptual and procedural understanding of vector components using both open ended problems and multiple choice items designed based on research instruments used in physics education research. The results showed that a number of students produce errors and revealed alternative conceptions especially when asked to draw graphical form of vector components. It indicated that most of them did not develop a strong foundation of understanding in vector components and could not apply those concepts to such problems with physical context. Based on the findings, we designed a worksheet to enhance the students’ conceptual understanding in vector components. The worksheet is composed of three parts which help students to construct their own understanding of definition, graphical form, and magnitude of vector components. To validate the worksheet, focus group discussions of 3 and 10 graduate students (science in-service teachers) had been conducted. The modified worksheet was then distributed to 41 grade 9 students in a science class. The students spent approximately 50 minutes to complete the worksheet. They sketched and measured vectors and its components and compared with the trigonometry ratio to condense the concepts of vector components. After completing the worksheet, their conceptual model had been verified. 83% of them constructed the correct model of vector components.

Faded-example as a Tool to Acquire and Automate Mathematics Knowledge

NASA Astrophysics Data System (ADS)

Retnowati, E.

2017-04-01

Students themselves accomplish Knowledge acquisition and automation. The teacher plays a role as the facilitator by creating mathematics tasks that assist students in building knowledge efficiently and effectively. Cognitive load caused by learning material presented by teachers should be considered as a critical factor. While the intrinsic cognitive load is related to the degree of complexity of the material learning ones can handle, the extraneous cognitive load is directly caused by how the material is presented. Strategies to present a learning material in computational learning domains like mathematics are a namely worked example (fully-guided task) or problem-solving (discovery task with no guidance). According to the empirical evidence, learning based on problem-solving may cause high-extraneous cognitive load for students who have limited prior knowledge, conversely learn based on worked example may cause high-extraneous cognitive load for students who have mastered the knowledge base. An alternative is a faded example consisting of the partly-completed task. Learning from faded-example can facilitate students who already acquire some knowledge about the to-be-learned material but still need more practice to automate the knowledge further. This instructional strategy provides a smooth transition from a fully-guided into an independent problem solver. Designs of faded examples for learning trigonometry are discussed.

Quantum Mechanics for Everybody: An autonomous MOOC on EdX for nonscientists

NASA Astrophysics Data System (ADS)

Freericks, James; Cutler, Dylan; Vieira-Barbosa, Lucas

2017-01-01

We have launched a MOOC for nonscientists that teaches quantum mechanics using the Feynman methodology as outlined in his QED book and in a similar book by Daniel Styer. Using a combination of videos, voice-over powerpoint animations, computer simulations and interactive tutorials, we teach the fundamentals of quantum mechanics employing a minimum of math (high school algebra, square roots, and a little trigonometry) but going into detail on a number of complex quantum ideas. We begin with the Stern-Gerlach experiment, including delayed choice and Bell's inequality variants. Then we focus on light developing the quantum theory for partial reflection and diffraction. At this point we demonstrate the complexity of quantum physics by showing how watched and unwatched two-slit experiments behave differently and how quantum particles interfere. The four week course ends with advanced topics in light where we cover the idea of an interaction free measurement, the quantum Zeno effect and indistinguishable particles via the Hong-Ou-Mandel experiment. We hope this MOOC will reach thousands of students interesting in learning quantum mechanics without any dumbing down or the need to learn complex math. It can also be used with undergraduates to help with conceptual understanding. Funded by the National Science Foundation under grants numbered PHY-1620555 and PHY-1314295 and by Georgetown University.

Johannes Kepler - And the New Astronomy

NASA Astrophysics Data System (ADS)

Voelkel, James R.

1999-11-01

Johannes Kepler (1571-1630) is remembered as one of the greatest medieval astronomers in the tradition of Copernicus and Galileo, a man who made major contributions to physics, astronomy, and mathematics. Born in Germany and trained as a theologian, Kepler did not hesitate to challenge church doctrine by supporting the iconoclastic theory of a Sun-centered solar system. As Imperial Mathematician to the Holy Roman Emperor, he conducted careful observations of the night sky, which led to his discovery of the three Laws of Planetary Motion and the orbit of Mars. He also devised the Rudolphine Tables on planetary movements, and made key improvements to the telescope. Voelkel vividly describes the scientific achievements, providing enough background in physics and trigonometry so even beginners can enjoy this book. The author also gives us a captivating account of Kepler's tumultuous life, plagued by misery, disease, and fervent religious prosecution by the Catholic Church.Oxford Portraits in Science is an ongoing series of scientific biographies for young adults. Written by top scholars and writers, each biography examines the personality of its subject as well as the thought process leading to his or her discoveries. These illustrated biographies combine accessible technical information with compelling personal stories to portray the scientists whose work has shaped our understanding of the natural world.

Prediction of the distance from skin to epidural space for low-thoracic epidural catheter insertion by computed tomography.

PubMed

Kao, M C; Tsai, S K; Chang, W K; Liu, H T; Hsieh, Y C; Hu, J S; Mok, M S

2004-02-01

It may be clinically useful to predict the depth of the epidural space. To investigate the accuracy of preoperative abdominal computed tomography (CT) in prediction of the distance for low-thoracic epidural insertion, a single group observational study was conducted in 30 male patients undergoing elective major abdominal surgery requiring epidural analgesia for postoperative pain relief. Using the paramedian approach, low-thoracic epidural insertion at T10-11 interspace was performed with a standardized procedure to obtain an actual insertion length (AIL). According to the principles of trigonometry, an estimated insertion length (EIL) was calculated as 1.26 times the distance from skin to epidural space measured from the preoperative abdominal CT. The mean (SD) EIL and AIL were 5.5 (0.7) and 5.1 (0.6) cm, respectively, with a significant correlation (r=0.899, P<0.01). The EIL tended to have a higher value than the AIL (0.4 (0.3) cm). There were significant correlations of both EIL and AIL with weight (P<0.01), BMI (P<0.01), and body fat percentage (P<0.01), but not with height (P>0.05). We conclude that the preoperative abdominal CT is helpful in prediction of the distance for low-thoracic epidural insertion using the paramedian approach.

Spatial filtering and spatial primitives in early vision: an explanation of the Zöllner-Judd class of geometrical illusion.

PubMed

Morgan, M J; Casco, C

1990-10-22

The apparent length and orientation of short lines is altered when they abut against oblique lines (the Zöllner and Judd illusions). Here we present evidence that the length and orientation biases are geometrically related and probably depend upon the same underlying mechanism. Measurements were done with an 'H' figure, in which the apparent length and orientation of the cross-bar was assessed by the method of adjustment while the orientation of the outer flanking lines was varied. When the flanking lines are oblique the apparent length of the central line is reduced and its orientation is shifted so that it appears more nearly at right-angles to the obliques than is in fact the case. Measurements of the orientation and length effects were made in three observers, over a range of flanking-line angles (90, 63, 45, 34 and 27 deg) and central line lengths (9, 17, 33 and 67 arc min). The biases increased with the tilt of the flanking-lines, and decreased with central line length. The extent of the length bias could be accurately predicted from the angular shift by simple trigonometry. We describe physiological and computational models to account for the relation between the orientation and length biases.

A new isocentric technique for exact geometric matching in the radiotherapy of the breast and ipsilateral supraclavicular fossa using dual asymmetric jaws.

PubMed

Romeo, Nando

2012-10-01

In some clinical situations breast or chest wall radiotherapy for cancer is given in association with supraclavicular fossa irradiation. Often the treatment is delivered by two tangential fields to the breast or chest wall and an anterior field that irradiates the supraclavicular region. The tissue between the breast or chest wall and the supraclavicular region may be under or overdosed, because of the junction between the two tangential fields and the anterior field. To present a new isocentric technique for exact geometric matching between the two tangential fields and the anterior field. Patients are positioned with both arms raised. Using three-dimensional trigonometry, two half-fields, with isocenter between the breast and the supraclavicular region, are easily matched. The tangential fields have a collimator rotation to protect the lung without additional shielding. The correct gantry, collimator and couch positions are defined for the anterior field to match the tangential fields. A general formula for exact geometric matching in radiotherapy of the breast and supraclavicular fossa is presented. The method does not require additional shielding to eliminate divergence other than the four independent jaws. The result is simple to implement in modern delivery facilities. Copyright © 2011 Associazione Italiana di Fisica Medica. Published by Elsevier Ltd. All rights reserved.

Linear measurements in 2-dimensional pelvic floor imaging: the impact of slice tilt angles on measurement reproducibility.

PubMed

Hoyte, L; Ratiu, P

2001-09-01

Magnetic resonance imaging techniques have improved the study of female pelvic dysfunction. However, disagreements between magnetic resonance measurements and their derived 3-dimensional reconstructions were noted. We tested the hypothesis that these discrepancies stemmed from variations in magnetic resonance acquisition angle. Images from the pelvis of the Visible Human Female (a thinly sliced cadaveric image data set) were obtained. Slices in the axial plane were rotated around pivot points in the pelvis to yield a set of similar-appearing para-axial images. A parameter that described the maximum anterior-posterior dimension of the levator hiatus was defined. This levator hiatus parameter was measured on all of the rotated images and compared with an expected value that was calculated from trigonometry. The levator hiatus was also measured on a group of similar-appearing slices rotated slightly around a defined point. In 1 group of slices, expected levator hiatus variation was 1.5 to 6.1%, whereas measured variation was 4% to 15%. Among the similar-appearing rotated slices, 4.8% to 16.0% variations were seen in the levator hiatus. Identical measurements made on radiologic images can vary widely. Slice acquisition must be standardized to avoid errors in data comparison.

Predictive formula for the length of tibial tunnel in anterior crucitate ligament reconstruction.

PubMed

Chernchujit, Bancha; Barthel, Thomas

2009-12-01

The anterior cruciate ligament (ACL) reconstruction using bone-patellar tendon bone graft is a common procedure in orthopedics. One challenging problem found is a graft-tunnel mismatch. Previous studies have reported the mathematic formula to predict the tibial angle length and angle to avoid graft-tunnel mismatch but these formulas have shown limited predictability. To propose a predictive formula for the length of tibial tunnel and to examine its predictability. Thirty six patients (26 males, 14 females) with ACL injury were included in this study. The preoperativemedial proximal tibial angle was measured. Intraoperatively, the tibial tunnel length and tibial entry point were measured. The postoperative coronal and saggital angle of tibial tunnel were measured from knee radiograph. The data were analysed by using trigonometry correlation and formulate the predictive formula of tibial tunnel length. We found that tibial tunnel length (T) has trigonometric correlation between the location of tibial tunnel entry point (w), coronal angle of tibial tunnel (b), saggital angle of tibial tunnel (a) and the medial proximal tibial slope (c) by using this formula T = Wcos(c)tan(b)/sin(a) This proposed predictive formula can well predict the length of the tibial tunnel at preoperative period to avoid graft-tunnel mismatch.

Digital image analysis: improving accuracy and reproducibility of radiographic measurement.

PubMed

Bould, M; Barnard, S; Learmonth, I D; Cunningham, J L; Hardy, J R

1999-07-01

To assess the accuracy and reproducibility of a digital image analyser and the human eye, in measuring radiographic dimensions. We experimentally compared radiographic measurement using either an image analyser system or the human eye with digital caliper. The assessment of total hip arthroplasty wear from radiographs relies on both the accuracy of radiographic images and the accuracy of radiographic measurement. Radiographs were taken of a slip gauge (30+/-0.00036 mm) and slip gauge with a femoral stem. The projected dimensions of the radiographic images were calculated by trigonometry. The radiographic dimensions were then measured by blinded observers using both techniques. For a single radiograph, the human eye was accurate to 0.26 mm and reproducible to +/-0.1 mm. In comparison the digital image analyser system was accurate to 0.01 mm with a reproducibility of +/-0.08 mm. In an arthroplasty model, where the dimensions of an object were corrected for magnification by the known dimensions of a femoral head, the human eye was accurate to 0.19 mm, whereas the image analyser system was accurate to 0.04 mm. The digital image analysis system is up to 20 times more accurate than the human eye, and in an arthroplasty model the accuracy of measurement increases four-fold. We believe such image analysis may allow more accurate and reproducible measurement of wear from standard follow-up radiographs.

Comparing the reliability of a trigonometric technique to goniometry and inclinometry in measuring ankle dorsiflexion.

PubMed

Sidaway, Ben; Euloth, Tracey; Caron, Heather; Piskura, Matthew; Clancy, Jessica; Aide, Alyson

2012-07-01

The purpose of this study was to compare the reliability of three previously used techniques for the measurement of ankle dorsiflexion ROM, open-chained goniometry, closed-chained goniometry, and inclinometry, to a novel trigonometric technique. Twenty-one physiotherapy students used four techniques (open-chained goniometry, closed-chained goniometry, inclinometry, and trigonometry) to assess dorsiflexion range of motion in 24 healthy volunteers. All student raters underwent training to establish competence in the four techniques. Raters then measured dorsiflexion with a randomly assigned measuring technique four times over two sessions, one week apart. Data were analyzed using a technique by session analysis of variance, technique measurement variability being the primary index of reliability. Comparisons were also made between the measurements derived from the four techniques and those obtained from a computerized video analysis system. Analysis of the rater measurement variability around the technique means revealed significant differences between techniques with the least variation being found in the trigonometric technique. Significant differences were also found between the technique means but no differences between sessions were evident. The trigonometric technique produced mean ROMs closest in value to those derived from computer analysis. Application of the trigonometric technique resulted in the least variability in measurement across raters and consequently should be considered for use when changes in dorsiflexion ROM need to be reliably assessed. Copyright © 2012 Elsevier B.V. All rights reserved.

Cardiac MR in robotic heart surgery for preoperative identification of the target vessel and precise port placement--a theoretical model.

PubMed

Bergmann, P; Huber, S; Segl, H; Maechler, H; Reiter, U; Reiter, G; Rienmueller, R; Oberwalder, P; Rigler, B

2003-08-01

The identification of the ideal anastomosis site and the proper port placement are critical for the success of closed-chest robotic surgery. We investigated a new systematic procedure for precise port placement for TECABs. We used trigonometry and a human thoracic model to determine the optimal working angles between anastomotic plane, instruments, and endoscope. We then applied the results to seven human subjects as follows: 1. A navigation grid was located extrathoracically before cardiac MR examination. 2. The ideal anastomosis site was defined with the MR. Intrathoracic distances and angles were computed with cardiac MR software and projected onto the thorax. 3. The ideal port placement points were marked on the thorax. The optimal working angle between endoscope and instruments was 35 degrees. 0 degrees and 90 degrees angles were associated with a significant reduction in visualization, technical ease, quality and anastomosis time. The course of the LAD was identified in all seven volunteers with MR. Mean deviation of the endoscope port from the medioclavicular line was 4.3+/-2.1 cm and of the instrument ports from the anterior axillary line 8.4+/-2.4 cm. Cardiac MR in combination with the navigation grid proved suitable for the visualization of coronary vessels for individually calculating port placement points on the thorax.

Geometric analysis of the V-Y advancement flap and its clinical applications.

PubMed

Andrades, Patricio R; Calderon, Wilfredo; Leniz, Patricio; Bartel, German; Danilla, Stefan; Benitez, Susana

2005-05-01

Geometry is fundamental in the comprehension of local flap design. The purpose of this study was to discuss the differences between the V-Y advancement flap and other local flaps, understand its geometry, and analyze its clinical applications. The analysis was based on qualitative measurements of an injury, taking into consideration the following dimensions: largest diameter, shortest diameter, and depth. Standardization of the flap design consisted of directing its advancement over the shortest diameter and making the V base match the size of the largest diameter. The flap was analyzed in two planes: the horizontal plane includes the V-Y design and the vertical plane includes the flap pedicle. The height of the flap can be obtained by simple trigonometry, taking into consideration the largest diameter and alpha angle in the horizontal plane. In the vertical plane, where the pedicle and pivot plane are positioned, for known shortest diameter and depth, the final depth of the pivot plane can be calculated using Pythagoras' principles. This analysis was applied to 25 patients with adequate skin coverage at follow-up. A correction factor was added to reduce the overdeepening of the vertical plane calculations. The final concepts for clinical application in the classic deep pedicle V-Y flap design are to calculate the length of the V by modifying the alpha angle and to move the pivot plane deeper to accomplish optimal flap movement. Using these principles, tension-free closure of the Y and appropriate advancement of the flap are obtained.

History of Astronomy in Portugal: Theories, Institutions and Practices

NASA Astrophysics Data System (ADS)

Saraiva, Luis

2014-01-01

In Portugal, throughout its history, astronomy was developed in the context of the mathematical sciences. During the times of Portugal's Maritime Discoveries, astronomical navigation was based on spherical trigonometry, and therefore it was the mathematicians who taught astronomy to the pilots. During the 17th century, basic notions of astronomy were taught in mathematical courses in the University and in the main Jesuit colleges. This tradition continued in the 18th century, so it is no wonder that one of the most influent Portuguese astronomers during this period was the mathematician José Monteiro da Rocha. During the 19th century the new centres of science teaching, as the Polytechnic School in Lisbon, or the Polytechnic Academy in Oporto, developed astronomy teaching and research in the context of the mathematics subjects. The inheritors of these 19th century institutions, respectively the Faculties of Sciences of Lisbon and Oporto, upheld this tradition until the final decades of 20th century and continued to consider astronomy as a subject to be taught in their mathematics departments. This Meeting aims at outlining several perspectives on the history of astronomy in Portugal, particularly analysing its ties with mathematical sciences and astronomy applications. The Meeting is organised by the Museum of Science of the University of Lisbon (MCUL) with CMAF, CMUC, CMUP and the CIUHCT, and is included in CIM events. It is integrated in the commemorations of the International Year of Astronomy (IYA2009).

Can partial coherence interferometry be used to determine retinal shape?

PubMed

Atchison, David A; Charman, W Neil

2011-05-01

To determine likely errors in estimating retinal shape using partial coherence interferometric instruments when no allowance is made for optical distortion. Errors were estimated using Gullstrand no. 1 schematic eye and variants which included a 10 diopter (D) axial myopic eye, an emmetropic eye with a gradient-index lens, and a 10.9 D accommodating eye with a gradient-index lens. Performance was simulated for two commercial instruments, the IOLMaster (Carl Zeiss Meditec) and the Lenstar LS 900 (Haag-Streit AG). The incident beam was directed toward either the center of curvature of the anterior cornea (corneal-direction method) or the center of the entrance pupil (pupil-direction method). Simple trigonometry was used with the corneal intercept and the incident beam angle to estimate retinal contour. Conics were fitted to the estimated contours. The pupil-direction method gave estimates of retinal contour that were much too flat. The cornea-direction method gave similar results for IOLMaster and Lenstar approaches. The steepness of the retinal contour was slightly overestimated, the exact effects varying with the refractive error, gradient index, and accommodation. These theoretical results suggest that, for field angles ≤30°, partial coherence interferometric instruments are of use in estimating retinal shape by the corneal-direction method with the assumptions of a regular retinal shape and no optical distortion. It may be possible to improve on these estimates out to larger field angles by using optical modeling to correct for distortion.

A technique for treating local breast cancer using a single set-up point and asymmetric collimation.

PubMed

Rosenow, U F; Valentine, E S; Davis, L W

1990-07-01

Using both pairs of asymmetric jaws of a linear accelerator local-regional breast cancer may be treated from a single set-up point. This point is placed at the abutment of the supraclavicular fields with the medial and lateral tangential fields. Positioning the jaws to create a half-beam superiorly permits treatment of the supraclavicular field. Positioning both jaws asymmetrically at midline to define a single beam in the inferoanterior quadrant permits treatment of the breast from medial and lateral tangents. The highest possible matching accuracy between the supraclavicular and tangential fields is inherently provided by this technique. For treatment of all fields at 100 cm source to axis distance (SAD) the lateral placement and depth of the set-up point may be determined by simulation and simple trigonometry. We elaborate on the clinical procedure. For the technologists treatment of all fields from a single set-up point is simple and efficient. Since the tissue at the superior border of the tangential fields is generally firmer than in mid-breast, greater accuracy in day-to-day set-up is permitted. This technique eliminates the need for table angles even when tangential fields only are planned. Because of half-beam collimation the limit to the tangential field length is 20 cm. Means will be suggested to overcome this limitation in the few cases where it occurs. Another modification is suggested for linear accelerators with only one independent pair of jaws.

Radiographic evaluation of BFX acetabular component position in dogs.

PubMed

Renwick, Alasdair; Gemmill, Toby; Pink, Jonathan; Brodbelt, David; McKee, Malcolm

2011-07-01

To assess the reliability of radiographic measurement of angle of lateral opening (ALO) and angle of version of BFX acetabular cups. In vitro radiographic study. BFX cups (24, 28, and 32 mm). Total hip replacement constructs (cups, 17 mm femoral head and a #7 CFX stem) were mounted on an inclinometer. Ventrodorsal radiographs were obtained with ALO varying between 21° and 70° and inclination set at 0°, 10°, 20°, and 30°. Radiographs were randomized using a random sequence generator. Three observers blinded to the radiograph order assessed ALO using 3 methods: (1) an ellipse method based on trigonometry; (2) using a measurement from the center of the femoral head to the truncated surface of the cup; (3) by visual estimation using a reference chart. Version was measured by assessing the ventral edge of the truncated surface. ALO methods 2 and 3 were accurate and precise to within 10° and were significantly more accurate and precise than method 1 (P < .001). All methods were significantly less accurate with increasing inclination. Version measurement was accurate and precise to within 7° with 0-20° of inclination, but significantly less accurate with 30° of inclination. Methods 2 and 3, but not method 1, were sufficiently accurate and precise to be clinically useful. Version measurement was clinically useful when inclination was ≤ 20°. © Copyright 2011 by The American College of Veterinary Surgeons.

Piloting a Global Collaborative Experiment to Determine your Place on the Planet and the Circumference of the Earth

NASA Astrophysics Data System (ADS)

Solie, D. J.; Paniwozik, R. L.; Wallace, P.

2012-12-01

As part of the laboratory component in Bush Physics for the 21st Century, a distance delivered physics course geared toward rural and Indigenous students in Alaska, students determine their village location on earth from simple sun angle measurements at local-noon during the spring equinox. Students measure the length of the sun shadow cast by a rod mounted on a horizontal surface, over short time intervals on or near the spring equinox during mid-day. Local-noon occurs when the sun is the highest and its corresponding shadow the shortest. Local noon, when expressed in Universal Time, can be directly converted to the local longitude in degrees. Local latitude in degrees, is obtained from the local-noon shadow length on the spring equinox and simple trigonometry. As an added bonus, using data from different sites, students can collaborate to approximate the circumference of the earth from their measurements. In the spirit of Eratosthenes, students envision an earth-sized pie wedge cut from a polar great-circle where the curve of the wedge on the earth's surface is the North-South distance between two often road-less sites (determined using Google Earth, a map or a globe), and the angle of the wedge is the difference between the site latitudes. The earth's circumference is calculated from this wedge. In 2012 with the aim of including Indigenous groups from other regions of the planet, we expanded this experiment to include teams from Japan, Puerto Rico, American Samoa, and New Zealand. We present our results from this pilot year.

A Vectorial Model to Compute Terrain Parameters, Local and Remote Sheltering, Scattering and Albedo using TIN Domains for Hydrologic Modeling.

NASA Astrophysics Data System (ADS)

Moreno, H. A.; Ogden, F. L.; Steinke, R. C.; Alvarez, L. V.

2015-12-01

Triangulated Irregular Networks (TINs) are increasingly popular for terrain representation in high performance surface and hydrologic modeling by their skill to capture significant changes in surface forms such as topographical summits, slope breaks, ridges, valley floors, pits and cols. This work presents a methodology for estimating slope, aspect and the components of the incoming solar radiation by using a vectorial approach within a topocentric coordinate system by establishing geometric relations between groups of TIN elements and the sun position. A normal vector to the surface of each TIN element describes slope and aspect while spherical trigonometry allows computing a unit vector defining the position of the sun at each hour and DOY. Thus, a dot product determines the radiation flux at each TIN element. Remote shading is computed by scanning the projection of groups of TIN elements in the direction of the closest perpendicular plane to the sun vector. Sky view fractions are computed by a simplified scanning algorithm in prescribed directions and are useful to determine diffuse radiation. Finally, remote radiation scattering is computed from the sky view factor complementary functions for prescribed albedo values of the surrounding terrain only for significant angles above the horizon. This methodology represents an improvement on the current algorithms to compute terrain and radiation parameters on TINs in an efficient manner. All terrain features (e.g. slope, aspect, sky view factors and remote sheltering) can be pre-computed and stored for easy access for a subsequent ground surface or hydrologic simulation.

Mathematical Understanding of the Underprivileged Students through GeoGebra

NASA Astrophysics Data System (ADS)

Amam, A.; Fatimah, A. T.; Hartono, W.; Effendi, A.

2017-09-01

A student’s mathematical understanding in high school from poor families in the district of Ciamis is still low. After reviews the various literature and earlier research, consequently, researchers convince that learning mathematics with GeoGebra can help students improve for the better understanding. Our long-term goal of this research is to support the implementation of new curriculum, namely ICT-based learning mathematics. Another goal is to give a basic mastery skill regarding mathematics software to students from underprivileged families. Moreover, the specific objective of this study is to examine the students’ mathematical understanding from underprivileged families after the implementation of learning with GeoGebra. We use a quantitative comparative research method to determine differences in the mathematical understanding of students’ from underprivileged families before and after mathematics learning with GeoGebra. Accordingly, the students of senior high school from underprivileged family in Baregbeg, Ciamis district, are the population of this study. This research is using purposive sampling. The instrument is in the form of a test question, which is the test of mathematical understanding. Research results show that the mathematical understanding students’ from underprivileged families after the mathematics learning with GeoGebra becomes better than before. The novelty of this research is that students understand the material of trigonometry through the use of modules, aided by GeoGebra in learning activities. Thus, the understanding has an impact on improving students’ mathematical understanding. Students also master the use of GeoGebra Software. Implementing these two things will be very useful for the next lesson.

Effects of minute misregistrations of prefabricated markers for image-guided dental implant surgery: an analytical evaluation.

PubMed

Rußig, Lorenz L; Schulze, Ralf K W

2013-12-01

The goal of the present study was to develop a theoretical analysis of errors in implant position, which can occur owing to minute registration errors of a reference marker in a cone beam computed tomography volume when inserting an implant with a surgical stent. A virtual dental-arch model was created using anatomic data derived from the literature. Basic trigonometry was used to compute effects of defined minute registration errors of only voxel size. The errors occurring at the implant's neck and apex both in horizontal as in vertical direction were computed for mean ±95%-confidence intervals of jaw width and length and typical implant lengths (8, 10 and 12 mm). Largest errors occur in vertical direction for larger voxel sizes and for greater arch dimensions. For a 10 mm implant in the frontal region, these can amount to a mean of 0.716 mm (range: 0.201-1.533 mm). Horizontal errors at the neck are negligible, with a mean overall deviation of 0.009 mm (range: 0.001-0.034 mm). Errors increase with distance to the registration marker and voxel size and are affected by implant length. Our study shows that minute and realistic errors occurring in the automated registration of a reference object have an impact on the implant's position and angulation. These errors occur in the fundamental initial step in the long planning chain; thus, they are critical and should be made aware to users of these systems. © 2012 John Wiley & Sons A/S.

Finding the optimal lengths for three branches at a junction.

PubMed

Woldenberg, M J; Horsfield, K

1983-09-21

This paper presents an exact analytical solution to the problem of locating the junction point between three branches so that the sum of the total costs of the branches is minimized. When the cost per unit length of each branch is known the angles between each pair of branches can be deduced following reasoning first introduced to biology by Murray. Assuming the outer ends of each branch are fixed, the location of the junction and the length of each branch are then deduced using plane geometry and trigonometry. The model has applications in determining the optimal cost of a branch or branches at a junction. Comparing the optimal to the actual cost of a junction is a new way to compare cost models for goodness of fit to actual junction geometry. It is an unambiguous measure and is superior to comparing observed and optimal angles between each daughter and the parent branch. We present data for 199 junctions in the pulmonary arteries of two human lungs. For the branches at each junction we calculated the best fitting value of x from the relationship that flow alpha (radius)x. We found that the value of x determined whether a junction was best fitted by a surface, volume, drag or power minimization model. While economy of explanation casts doubt that four models operate simultaneously, we found that optimality may still operate, since the angle to the major daughter is less than the angle to the minor daughter. Perhaps optimality combined with a space filling branching pattern governs the branching geometry of the pulmonary artery.

Muscle length alters geometry of arterioles and venules in hamster retractor.

PubMed

Nakao, M; Segal, S S

1995-01-01

We investigated how changes in muscle length (Lm) would alter the geometry of arterioles and venules and whether such an effect would depend on the orientation of microvessels to muscle fibers. The parallel-fibered retractor muscle of anesthetized (pentobarbital sodium, 65 mg/kg) male hamsters (n = 20, 105 +/- 4 g) was exposed and irrigated with physiological saline solution (pH 7.4; 35 degrees C). Sarcomere length (Ls) was measured at x2,400 magnification after topical application (3 min, 10(-5) M) of a fluorescent dye [4-(4-diethylaminostyryl)-N-methylpyridinium iodide]. In vivo Ls at resting Lm (i.e., at Lm = 100%) was 3.00 +/- 0.02 microns. The origin and insertion of the retractor were cut, and the muscle was reflected dorsally while the circulation arising from the ventral surface was preserved. Polystyrene "tendons" were glued to each end of the muscle to control Lm, which was varied in 10% increments from 80 to 130% of in situ Lm; Ls increased linearly (r2 = 0.82) from 2.58 +/- 0.03 to 3.89 +/- 0.07 microns, respectively. Arteriole and venule branches and the centerline of "Y" bifurcations were classified based on orientation angles (theta) with respect to muscle fibers at Lm = 100%; three categories were defined using trigonometry (detailed in the APPENDIX) based on microvessel behavior during changes in Lm: parallel (P), 0 degree < or = theta < or = 32.6 degrees; intermediate (I), 32.6 degrees < theta < 59.4 degrees; and normal (N), 59.4 degrees < or = theta < or = 90 degrees.(ABSTRACT TRUNCATED AT 250 WORDS)

Validation, repeatability and reproducibility of a noninvasive instrument for measuring thoracic and lumbar curvature of the spine in the sagittal plane.

PubMed

Chaise, Fabiana O; Candotti, Cláudia T; Torre, Marcelo L; Furlanetto, Tássia S; Pelinson, Patricia P T; Loss, Jefferson F

2011-01-01

The need for early identification of postural abnormalities without exposing patients to constant radiation has stimulated the development of instruments aiming to measure the spinal curvatures. To verify the validity, repeatability and reproducibility of angular measures of sagittal curvatures of the spine obtained using an adapted arcometer, by comparing them with Cobb angles of the respective curvatures obtained by using X-rays. 52 participants were submitted to two procedures designed to evaluate the thoracic and lumbar curvatures: (1) X-ray examination from which the Cobb angles (CA) of both curvatures were obtained, and (2) measuring the angles with the arcometer (AA). Two evaluators collected the data using the arcometer, with the rods placed at T1, T12, L1 and L5 spinous processes levels in a way as to permit linear measurements which, with aid of trigonometry, supplied the AA. There was a very strong and significant correlation between AA and CA (r=0.94; p<0.01), with no-significant difference (p=0.32), for the thoracic curvature. There was a strong and significant correlation for the lumbar curvature (r=0.71; p<0.01) between AA and CA, with no-significant difference (p=0.30). There is a very strong correlation between intra-evaluator and inter-evaluator AA. It was possible to quantify reliably the thoracic and lumbar curvatures with the arcometer and it can thus be considered valid and reliable and for use in evaluating spinal curvatures in the sagittal plane.

A novel and fast method for proton range verification using a step wedge and 2D scintillator.

PubMed

Shen, Jiajian; Allred, Bryce C; Robertson, Daniel G; Liu, Wei; Sio, Terence T; Remmes, Nicholas B; Keole, Sameer R; Bues, Martin

2017-09-01

To implement and evaluate a novel and fast method for proton range verification by using a planar scintillator and step wedge. A homogenous proton pencil beam plan with 35 energies was designed and delivered to a 2D flat scintillator with a step wedge. The measurement was repeated 15 times (3 different days, 5 times per day). The scintillator image was smoothed, the Bragg peak and distal fall off regions were fitted by an analytical equation, and the proton range was calculated using simple trigonometry. The accuracy of this method was verified by comparing the measured ranges to those obtained using an ionization chamber and a scanning water tank, the gold standard. The reproducibility was evaluated by comparing the ranges over 15 repeated measurements. The sensitivity was evaluated by delivering to same beam to the system with a film inserted under the wedge. The range accuracy of all 35 proton energies measured over 3 days was within 0.2 mm. The reproducibility in 15 repeated measurements for all 35 proton ranges was ±0.045 mm. The sensitivity to range variation is 0.1 mm for the worst case. This efficient procedure permits measurement of 35 proton ranges in less than 3 min. The automated data processing produces results immediately. The setup of this system took less than 5 min. The time saving by this new method is about two orders of magnitude when compared with the time for water tank range measurements. A novel method using a scintillator with a step wedge to measure the proton range was implemented and evaluated. This novel method is fast and sensitive, and the proton range measured by this method was accurate and highly reproducible. © 2017 American Association of Physicists in Medicine.

A study on making a Honsang using the star catalogue from 『Seong Gyeong』

NASA Astrophysics Data System (ADS)

Ham, Seon Young; Kim, Sang Hyuk; Lee, Yong Sam

2016-01-01

The first record of Honsang (Celestial globe) was found in 『Sejong Sillok』 in Korea. Since then, there were records that Honsang was restored during the reign of King Jungjong and King Myungjong, and then restored again in the reign of King Seonjo. The only existing Honsang was made by Yi Hwang (1501-1570) in the 16th century for education of his followers. After then, Hong Dae Yong's (1731-1783) Honsangui, which was made in 18th century, had been passed down only through the literature. The constellations in Honsang and the scale system of each ring changed after 17th century when Western science began to affect Joseon dynasty. Since that time, the constellations, realized on Honsang globe, changed from constellations in the old method to ones in the new method. Furthermore, the scale system of rings on Honsang was changed from 365.25 Do, Jucheondo (Celestial globe circumference), to 360°. In this study, Honsang with constellations in the new method was made using star catalogue from 『Seong Gyeong』 published in 1861, which represented the constellations in the new method of Joseon dynasty. In order to realized the constellations from the star catalogue in 『Seong Gyeong』 on Honsang globe, the plane star chart and circular star chart of the area near the South and North Poles were drawn using spherical trigonometry. Using these star charts, the constellations in whole sky including stars near the South Pole were realized on Honsang globe. Also, equatorial coordinates and ecliptic coordinates were realized on Honsang globe simultaneously, and scales of Honsang's rings were marked as 360°.

Lectures on Selected Topics in Mathematical Physics: Elliptic Functions and Elliptic Integrals

NASA Astrophysics Data System (ADS)

Schwalm, William A.

2015-12-01

This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.

In vivo fascicle length measurements via B-mode ultrasound imaging with single vs dual transducer arrangements.

PubMed

Brennan, Scott F; Cresswell, Andrew G; Farris, Dominic J; Lichtwark, Glen A

2017-11-07

Ultrasonography is a useful technique to study muscle contractions in vivo, however larger muscles like vastus lateralis may be difficult to visualise with smaller, commonly used transducers. Fascicle length is often estimated using linear trigonometry to extrapolate fascicle length to regions where the fascicle is not visible. However, this approach has not been compared to measurements made with a larger field of view for dynamic muscle contractions. Here we compared two different single-transducer extrapolation methods to measure VL muscle fascicle length to a direct measurement made using two synchronised, in-series transducers. The first method used pennation angle and muscle thickness to extrapolate fascicle length outside the image (extrapolate method). The second method determined fascicle length based on the extrapolated intercept between a fascicle and the aponeurosis (intercept method). Nine participants performed maximal effort, isometric, knee extension contractions on a dynamometer at 10° increments from 50 to 100° of knee flexion. Fascicle length and torque were simultaneously recorded for offline analysis. The dual transducer method showed similar patterns of fascicle length change (overall mean coefficient of multiple correlation was 0.76 and 0.71 compared to extrapolate and intercept methods respectively), but reached different absolute lengths during the contractions. This had the effect of producing force-length curves of the same shape, but each curve was shifted in terms of absolute length. We concluded that dual transducers are beneficial for studies that examine absolute fascicle lengths, whereas either of the single transducer methods may produce similar results for normalised length changes, and repeated measures experimental designs. Copyright © 2017 Elsevier Ltd. All rights reserved.

Irregular-Mesh Terrain Analysis and Incident Solar Radiation for Continuous Hydrologic Modeling in Mountain Watersheds

NASA Astrophysics Data System (ADS)

Moreno, H. A.; Ogden, F. L.; Alvarez, L. V.

2016-12-01

This research work presents a methodology for estimating terrain slope degree, aspect (slope orientation) and total incoming solar radiation from Triangular Irregular Network (TIN) terrain models. The algorithm accounts for self shading and cast shadows, sky view fractions for diffuse radiation, remote albedo and atmospheric backscattering, by using a vectorial approach within a topocentric coordinate system and establishing geometric relations between groups of TIN elements and the sun position. A normal vector to the surface of each TIN element describes slope and aspect while spherical trigonometry allows computingunit vector defining the position of the sun at each hour and day of the year. Thus, a dot product determines the radiation flux at each TIN element. Cast shadows are computed by scanning the projection of groups of TIN elements in the direction of the closest perpendicular plane to the sun vector only in the visible horizon range. Sky view fractions are computed by a simplified scanning algorithm from the highest to the lowest triangles along prescribed directions and visible distances, useful to determine diffuse radiation. Finally, remotealbedo is computed from the sky view fraction complementary functions for prescribed albedo values of the surrounding terrain only for significant angles above the horizon. The sensitivity of the different radiative components is tested a in a moutainuous watershed in Wyoming, to seasonal changes in weather and surrounding albedo (snow). This methodology represents an improvement on the current algorithms to compute terrain and radiation values on triangular-based models in an accurate and efficient manner. All terrain-related features (e.g. slope, aspect, sky view fraction) can be pre-computed and stored for easy access for a subsequent, progressive-in-time, numerical simulation.

Tensile properties of the transverse carpal ligament and carpal tunnel complex.

PubMed

Ugbolue, Ukadike C; Gislason, Magnus K; Carter, Mark; Fogg, Quentin A; Riches, Philip E; Rowe, Philip J

2015-08-01

A new sophisticated method that uses video analysis techniques together with a Maillon Rapide Delta to determine the tensile properties of the transverse carpal ligament-carpal tunnel complex has been developed. Six embalmed cadaveric specimens amputated at the mid-forearm and aged (mean (SD)): 82 (6.29) years were tested. The six hands were from three males (four hands) and one female (two hands). Using trigonometry and geometry the elongation and strain of the transverse carpal ligament and carpal arch were calculated. The cross-sectional area of the transverse carpal ligament was determined. Tensile properties of the transverse carpal ligament-carpal tunnel complex and Load-Displacement data were also obtained. Descriptive statistics, one-way ANOVA together with a post-hoc analysis (Tukey) and t-tests were incorporated. A transverse carpal ligament-carpal tunnel complex novel testing method has been developed. The results suggest that there were no significant differences between the original transverse carpal ligament width and transverse carpal ligament at peak elongation (P=0.108). There were significant differences between the original carpal arch width and carpal arch width at peak elongation (P=0.002). The transverse carpal ligament failed either at the mid-substance or at their bony attachments. At maximum deformation the peak load and maximum transverse carpal ligament displacements ranged from 285.74N to 1369.66N and 7.09mm to 18.55mm respectively. The transverse carpal ligament cross-sectional area mean (SD) was 27.21 (3.41)mm(2). Using this method the results provide useful biomechanical information and data about the tensile properties of the transverse carpal ligament-carpal tunnel complex. Copyright © 2015 Elsevier Ltd. All rights reserved.

Minority Undergraduate Training for Energy-Related Careers (MUTEC)

DOE Office of Scientific and Technical Information (OSTI.GOV)

Levy, C.; Yih, T.C.; Ebadian, M.A.

1995-12-01

OAK-B135 Minority Undergraduate Training for Energy-Related Careers (MUTEC). First, all the co-investigators would like to thank the Department of Energy's Minority Impact Office for awarding FIU with the MUTEC grant for the past five years. We believe it has made a difference, especially in the creation of a new, streamlined curriculum that began with the Mechanical Engineering Program and has now become college wide. Second, we have given 774 students an introduction to engineering, something that did not exist 3 years ago. Third, we have given FLAME the opportunity to participate in this program through the equivalent introduction to engineeringmore » course. Over 150 of those students have participated and have a 100% record of completing the program once, they start. Over 80% of those students have gone on to college. Fourth, we have aided 32 undergraduates continue in their engineering studies. Of those half have already graduated, and half of those have gone on to graduate school. One of these graduate school students has graduated with an MSME and another has won an NSF Scholarship. Fifth, we have created a bank of 51 2-hour tapes in 10 science and engineering science areas and covered the spectrum of math courses from geometry/trigonometry to differential equations. Sixth, we have created two examinations for use in preparation for entry into the engineering programs and in preparation for the EIT. Seventh, we have created a streamlined curriculum and four options, two of which are energy related. From these points, we believe that the program was very successful and for that we wish to thank the Department of Energy and specifically Ms. Estela Romo for her unwavering support.« less

Perspective distortion in craniofacial superimposition: Logarithmic decay curves mapped mathematically and by practical experiment.

PubMed

Stephan, Carl N

2015-12-01

The superimposition of a face photograph with that of a skull for identification purposes necessitates the use of comparable photographic parameters between the two image acquisition sessions, so that differences in optics and consequent recording of images does not thwart the morphological analysis. Widely divergent, but published, speculations about the thresholds at which perspective distortion becomes negligible (0.5 to >13.5 m) must be resolved and perspective distortion (PD) relationships quantified across their full range to judge tolerance levels, and the suitability of commonly employed contemporary equipment (e.g., 1 m photographic copy-stands). Herein, basic trigonometry is employed to map PD for two same sized 179 mm linear lengths - separated anteroposteriorly by 127 mm - as a function of subject-to-camera distance (SCD; 0.2-20 m). These lengths approximate basic craniofacial heights (e.g., tr-n) and widths (e.g., zy-zy), while the latter approximates facial depth (e.g., n-t). As anticipated, PD decayed in logarithmic and continuous manner with increasing SCD. At SCD of 12 m, the within-image PD was negligible (<1%). At <2.5 m SCD, it exceeded 5% and increased sharply as SCD decreased. Since life size images of skulls and faces are commonly employed for superimposition, a relative 1% perspective distortion difference is recommended as the ceiling standard for craniofacial comparison (translates into a ≤2 mm difference in physiognomical face height). Since superimposition depends on relative comparisons of a photographic pair (not one photograph), there is practically no scenario in superimposition casework where SCDs should be ignored and no single distance at which PD should be considered negligible (even if one image holds >12 m SCD). Copyright © 2015 Elsevier Ireland Ltd. All rights reserved.

Theoretical discrepancy between cage size and efficient tibial tuberosity advancement in dogs treated for cranial cruciate ligament rupture.

PubMed

Etchepareborde, S; Mills, J; Busoni, V; Brunel, L; Balligand, M

2011-01-01

To calculate the difference between the desired tibial tuberosity advancement (TTA) along the tibial plateau axis and the advancement truly achieved in that direction when cage size has been determined using the method of Montavon and colleagues. To measure the effect of this difference on the final patellar tendon-tibial plateau angle (PTA) in relation to the ideal 90°. Trigonometry was used to calculate the theoretical actual advancement of the tibial tuberosity in a direction parallel to the tibial plateau that would be achieved by the placement of a cage at the level of the tibial tuberosity in the osteotomy plane of the tibial crest. The same principle was used to calculate the size of the cage that would have been required to achieve the desired advancement. The effect of the difference between the desired advancement and the actual advancement achieved on the final PTA was calculated. For a given desired advancement, the greater the tibial plateau angle (TPA), the greater the difference between the desired advancement and the actual advancement achieved. The maximum discrepancy calculated was 5.8 mm for a 12 mm advancement in a case of extreme TPA (59°). When the TPA was less than 31°, the PTA was in the range of 90° to 95°. A discrepancy does exist between the desired tibial tuberosity advancement and the actual advancement in a direction parallel to the TPA, when the tibial tuberosity is not translated proximally. Although this has an influence on the final PTA, further studies are warranted to evaluate whether this is clinically significant.

Predicting translational deformity following opening-wedge osteotomy for lower limb realignment.

PubMed

Barksfield, Richard C; Monsell, Fergal P

2015-11-01

An opening-wedge osteotomy is well recognised for the management of limb deformity and requires an understanding of the principles of geometry. Translation at the osteotomy is needed when the osteotomy is performed away from the centre of rotation of angulation (CORA), but the amount of translation varies with the distance from the CORA. This translation enables proximal and distal axes on either side of the proposed osteotomy to realign. We have developed two experimental models to establish whether the amount of translation required (based on the translation deformity created) can be predicted based upon simple trigonometry. A predictive algorithm was derived where translational deformity was predicted as 2(tan α × d), where α represents 50 % of the desired angular correction, and d is the distance of the desired osteotomy site from the CORA. A simulated model was developed using TraumaCad online digital software suite (Brainlab AG, Germany). Osteotomies were simulated in the distal femur, proximal tibia and distal tibia for nine sets of lower limb scanograms at incremental distances from the CORA and the resulting translational deformity recorded. There was strong correlation between the distance of the osteotomy from the CORA and simulated translation deformity for distal femoral deformities (correlation coefficient 0.99, p < 0.0001), proximal tibial deformities (correlation coefficient 0.93-0.99, p < 0.0001) and distal tibial deformities (correlation coefficient 0.99, p < 0.0001). There was excellent agreement between the predictive algorithm and simulated translational deformity for all nine simulations (correlation coefficient 0.93-0.99, p < 0.0001). Translational deformity following corrective osteotomy for lower limb deformity can be anticipated and predicted based upon the angular correction and the distance between the planned osteotomy site and the CORA.

PubMed

Busard, H L L

1997-12-01

The twelfth century was a period of transmission and absorption of Arabic learning though it filtered outside of the Arabic world as early as the second half of the tenth century. In general, the lure of Spain began to act only in the twelfth century, and the active impulse toward the spread of Arabic mathematics came from beyond the Pyrenees and from men of diverse origins. The chief names are Adelard of Bath, Robert of Chester, Hermann of Carinthia and Gerard of Cremona. In this time the Latin world became acquainted with the Hindu numerals, the Arabic Algebra and Euclid'sElements. However, not only Spain, but also the Norman kingdom of southern Italy and Sicily occupies a position of peculiar importance, though the works of the translators did not become very influential. There were made direct translations from Greek into Latin. One had to wait a century more to obtain a translation from Greek into Latin of the chief Archimedean scientific and mathematical treatises by William of Moerbeke. In the thirteenth century Fibonacci and Jordanus Nemorarius stand at the threshold of European mathematics. Not only was Fibonacci the first to explain Arabic arithmetic, but his works, especially his later ones, contain many original ideas. Jordanus continued the Greco-Roman tradition rather than the Greco-Arabic one, but he did so with much independence. To Nicole Oresme (fourteenth century) was due a broadened view of proportionality, a geometric proof to determine the summation of convergent infinite series and the proof, evidently the first in the history of mathematics, that the harmonic series is divergent. The Configuration Doctrine was treated by Merton College authors and by Oresme. In the fifteenth century theDe triangulis omnimodis of Regiomontan, a systematic account of the methods for solving triangles, marked the rebirth of trigonometry.

Patient-specific rhytidectomy: finding the angle of maximal rejuvenation.

PubMed

Jacono, Andrew A; Ransom, Evan R

2012-09-01

Rhytidectomy is fundamentally an operation of tissue release and resuspension, although the manner and direction of suspension are subject to perpetual debate. The authors describe a method for identifying the angle of maximal rejuvenation during rhytidectomy and quantify the resulting angle and its relationship to patient age. Patients were prospectively enrolled; demographic data, history, and operative details were recorded. Rhytidectomies were performed by the senior author (AAJ). After complete elevation, the face-lift flap was rotated in a medially-based arc (0-90°) while attention was given to the submental area, jawline, and midface. The angle of maximal rejuvenation for each hemiface was identified as described, and the flap was resuspended. During redraping, measurements of vertical and horizontal skin excess were recorded in situ. The resulting angle of lift was then calculated for each hemiface using trigonometry. Symmetry between sides was determined, and the effect of patient age on this angle was assessed. Three hundred hemifaces were operated (147 women; 3 men). Mean age was 60 years (range, 37-80 years). Mean resulting angle for the cohort was 60° from horizontal (range, 46-77°). This was inversely correlated with patient age (r = -.3). Younger patients (<50 years, 64°) had a significantly more vertical angle than older patients (≥70 years, 56°; P < .0002). No significant intersubject difference was found between hemifaces (P = .53). The authors present a method for identifying the angle of maximal rejuvenation during rhytidectomy. This angle was more superior than posterior in all cases and is intimately related to patient age. Lasting results demand a detailed anatomical understanding and strict attention to the direction and degree of laxity.

A flexible and cost-effective compensation method for leveling using large-scale coordinate measuring machines and its application in aircraft digital assembly

NASA Astrophysics Data System (ADS)

Deng, Zhengping; Li, Shuanggao; Huang, Xiang

2018-06-01

In the assembly process of large-size aerospace products, the leveling and horizontal alignment of large components are essential prior to the installation of an inertial navigation system (INS) and the final quality inspection. In general, the inherent coordinate systems of large-scale coordinate measuring devices are not coincident with the geodetic horizontal system, and a dual-axis compensation system is commonly required for the measurement of difference in heights. These compensation systems are expensive and dedicated designs for different devices at present. Considering that a large-size assembly site usually needs more than one measuring device, a compensation approach which is versatile for different devices would be a more convenient and economic choice for manufacturers. In this paper, a flexible and cost-effective compensation method is proposed. Firstly, an auxiliary measuring device called a versatile compensation fixture (VCF) is designed, which mainly comprises reference points for coordinate transformation and a dual-axis inclinometer, and a kind of network tighten points (NTPs) are introduced and temporarily deployed in the large measuring space to further reduce transformation error. Secondly, the measuring principle of height difference is studied, based on coordinate transformation theory and trigonometry while considering the effects of earth curvature, and the coordinate transformation parameters are derived by least squares adjustment. Thirdly, the analytical solution of leveling uncertainty is analyzed, based on which the key parameters of the VCF and the proper deployment of NTPs are determined according to the leveling accuracy requirement. Furthermore, the proposed method is practically applied to the assembly of a large helicopter by developing an automatic leveling and alignment system. By measuring four NTPs, the leveling uncertainty (2σ) is reduced by 29.4% to about 0.12 mm, compared with that without NTPs.

STEPS at CSUN: Increasing Retention of Engineering and Physical Science Majors

NASA Astrophysics Data System (ADS)

Pedone, V. A.; Cadavid, A. C.; Horn, W.

2012-12-01

STEPS at CSUN seeks to increase the retention rate of first-time freshman in engineering, math, and physical science (STEM) majors from ~55% to 65%. About 40% of STEM first-time freshmen start in College Algebra because they do not take or do not pass the Mathematics Placement Test (MPT). This lengthens time to graduation, which contributes to dissatisfaction with major. STEPS at CSUN has made substantial changes to the administration of the MPT. Initial data show increases in the number of students who take the test and who place out of College Algebra, as well as increases in overall scores. STEPS at CSUN also funded the development of supplemental labs for Trigonometry and Calculus I and II, in partnership with similar labs created by the Math Department for College Algebra and Precalculus. These labs are open to all students, but are mandatory for at-risk students who have low scores on the MPT, low grades in the prerequisite course, or who failed the class the first time. Initial results are promising. Comparison of the grades of 46 Fall 2010 "at-risk" students without lab to those of 36 Fall 2011 students who enrolled in the supplementary lab show D-F grades decreased by 10% and A-B grades increased by 27%. A final retention strategy is aimed at students in the early stages of their majors. At CSUN the greatest loss of STEM majors occurs between sophomore-level and junior-level coursework because course difficulty increases and aspirations to potential careers weaken. The Summer Interdisciplinary Team Experience (SITE) is an intensive 3-week-long summer program that engages small teams of students from diverse STEM majors in faculty-mentored, team-based problem solving. This experience simulates professional work and creates strong bonds between students and between students and faculty mentors. The first two cohorts of students who have participated in SITE indicate that this experience has positively impacted their motivation to complete their STEM degree.

Predictors of Associate's Degree Completion in Engineering and Engineering Technologies

NASA Astrophysics Data System (ADS)

Reys-Nickel, Lynsey L.

The purpose of this ex post facto study was to describe completers and non-completers of associate's degree programs in engineering and engineering technologies and determine whether and to what extent completion in these programs is a function of selected student-related variables and institutional variables. Data from the 2004/2009 Beginning Postsecondary Students Longitudinal Study (BPS: 04/09) of associate's degree completers and non-completers in engineering and engineering technologies were accessed and analyzed through PowerStats, a web-based data analysis tool from National Center for Education Statistics (NCES). Descriptive data indicated that, proportionally, engineering and engineering technologies completers were mostly White, married, middle income, employed part-time, enrolled full-time, did not hold a high school diploma or certificate, completed Trigonometry/Algebra II, had a father who's highest education level was an associate's degree, but did not know their mother's highest level of education, completed remedial coursework, and started college with the goal of earning an associate's degree. While more males enrolled in the programs, males and females demonstrated similar completion rates, proportionally--with females showing a slightly higher percentage of completion. Results from the logistic regression further indicated that the variables significant to completion in associate's degree programs in engineering and engineering technologies were gender and enrollment size. Findings suggested that female students were more likely to earn the degree, and that the larger the institution, the more likely the student would become a completer. However, since a major limitation of the study was the small weighted sample size, the results of the study are inconclusive in terms of the extent to which the findings can be generalized to the population of students in associate's degree programs in engineering and engineering technologies. This study fills a gap in the literature of what is known about engineering and engineering technician students. It also contributes to the body of research on an understudied STEM educational and professional pathway, the associate's degree in engineering and engineering technologies.

Physics gains attraction?

NASA Astrophysics Data System (ADS)

2000-01-01

A recent report from the American Institute of Physics has indicated that high school enrolments in physics in the USA have reached their highest level since World War II. Figures for the last decade show an increase in the proportion of high school students taking physics from 20 to 28% (800Â 000 students now), according to Physics Today (October 1999, p 68). The report, Maintaining Momentum: High School Physics for a New Millennium , was based on a 1997 survey of high school physics teachers, the fourth such since the mid-1980s. One conclusion drawn by the report's authors was that a broader range of physics courses is now offered, with increased popularity of `conceptual' physics courses using little algebra or trigonometry over the last ten years. The proportion of students with the strongest maths abilities now taking advanced placement or second-year physics has doubled since 1987. In addition the physics appeal has been noted among high school girls, where the percentage taking physics has risen from 39 to 47% in the ten years to 1997. These female students do not, however, seem to extend their studies into advanced placements or even into teaching physics (women constitute just a quarter of high school teachers of the subject). Sadly the good news is outweighed by the fact that physics still registers the lowest enrolments of all the high school sciences - about half those in chemistry for example. Indeed only around 1% of high school students have taken two years of physics before they graduate, which represents a much lower proportion than in many European and Asian countries. The full report can be viewed at the AIP's statistics division's homepage: www.aip.org/statistics/trends/hstrends.htm whilst summaries of the document are available free from the AIP, Education and Employment Statistics Division, One Physics Ellipse, College Park, MD 20740, USA.

Software for determining the true displacement of faults

NASA Astrophysics Data System (ADS)

Nieto-Fuentes, R.; Nieto-Samaniego, Á. F.; Xu, S.-S.; Alaniz-Álvarez, S. A.

2014-03-01

One of the most important parameters of faults is the true (or net) displacement, which is measured by restoring two originally adjacent points, called “piercing points”, to their original positions. This measurement is not typically applicable because it is rare to observe piercing points in natural outcrops. Much more common is the measurement of the apparent displacement of a marker. Methods to calculate the true displacement of faults using descriptive geometry, trigonometry or vector algebra are common in the literature, and most of them solve a specific situation from a large amount of possible combinations of the fault parameters. True displacements are not routinely calculated because it is a tedious and tiring task, despite their importance and the relatively simple methodology. We believe that the solution is to develop software capable of performing this work. In a previous publication, our research group proposed a method to calculate the true displacement of faults by solving most combinations of fault parameters using simple trigonometric equations. The purpose of this contribution is to present a computer program for calculating the true displacement of faults. The input data are the dip of the fault; the pitch angles of the markers, slickenlines and observation lines; and the marker separation. To prevent the common difficulties involved in switching between operative systems, the software is developed using the Java programing language. The computer program could be used as a tool in education and will also be useful for the calculation of the true fault displacement in geological and engineering works. The application resolves the cases with known direction of net slip, which commonly is assumed parallel to the slickenlines. This assumption is not always valid and must be used with caution, because the slickenlines are formed during a step of the incremental displacement on the fault surface, whereas the net slip is related to the finite slip.

Grading apical vertebral rotation without a computed tomography scan: a clinically relevant system based on the radiographic appearance of bilateral pedicle screws.

PubMed

Upasani, Vidyadhar V; Chambers, Reid C; Dalal, Ali H; Shah, Suken A; Lehman, Ronald A; Newton, Peter O

2009-08-01

Bench-top and retrospective analysis to assess vertebral rotation based on the appearance of bilateral pedicle screws in patients with adolescent idiopathic scoliosis (AIS). To develop a clinically relevant radiographic grading system for evaluating postoperative thoracic apical vertebral rotation that would correlate with computed tomography (CT) measures of rotation. The 3-column vertebral body control provided by bilateral pedicle screws has enabled scoliosis surgeons to develop advanced techniques of direct vertebral derotation. Our ability to accurately quantify spinal deformity in the axial plane, however, continues to be limited. Trigonometry was used to define the relationship between the position of bilateral pedicle screws and vertebral rotation. This relationship was validated using digital photographs of a bench-top model. The mathematical relationships were then used to calculate vertebral rotation from standing postoperative, posteroanterior radiographs in AIS patients and correlated with postoperative CT measures of rotation. Fourteen digital photographs of the bench-top model were independently analyzed twice by 3 coauthors. The mathematically calculated degree of rotation was found to correlate significantly with the actual degree of rotation (r = 0.99; P < 0.001) and the intra- and interobserver reliability for these measurements were both excellent (kappa = 0.98 and kappa = 0.97, respectively). In the retrospective analysis of 17 AIS patients, the average absolute difference between the radiographic measurement of rotation and the CT measure was only 1.9 degrees +/- 2.0 degrees (r = 0.92; P < 0.001). Based on these correlations a simple radiographic grading system for postoperative apical vertebral rotation was developed. An accurate assessment of vertebral rotation can be performed radiographically, using screw lengths and screw tip-to-rod distances of bilateral segmental pedicle screws and a trigonometric calculation. These data support the use of a simple radiographic grading system to approximate apical vertebral rotation in AIS patients treated with bilateral apical pedicle screws.

SU-E-P-45: An Analytical Formula for Deriving Mechanical Iso-Center of Rotational Gantry Treatment Unit Rotational Gantry Treatment Unit

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ding, X; Bues, M

2015-06-15

Purpose: To present an analytical formula for deriving mechanical isocenter (MIC) of a rotational gantry treatment unit. The input data to the formula is obtained by a custom-made device. The formula has been implemented and used in an operational proton therapy facility since 2005. Methods: The custom made device consisted of 3 mutually perpendicular dial indicators and 5 clinometers, to obtain displacement data and gantry angle data simultaneously. During measurement, a steel sphere was affixed to the patient couch, and the device was attached to the snout rotating with the gantry. The displacement data and angle data were obtained simultaneouslymore » at angular increments of less than 1 degree. The analytical formula took the displacement and angle as input and derived the positions of dial indicator tips (DIT) position in room-fixed coordinate system. The formula derivation presupposes trigonometry and 3-dimentional coordinate transformations. Due to the symmetry properties of the defining equations, the DIT position can be solved for analytically without using mathematical approximations. We define the mean of all points in the DIT trajectory as the MIC. The formula was implemented in computer code, which has been employed during acceptance test, commissioning, as well as routine QA practice in an operational proton facility since 2005. Results: It took one minute for the custom-made device to acquire the measurement data for a full gantry rotation. The DIT trajectory and MIS are instantaneously available after the measurement. The MIC Result agrees well with vendor’s Result, which came from a different measurement setup, as well as different data analysis algorithm. Conclusion: An analytical formula for deriving mechanical isocenter was developed and validated. The formula is considered to be absolutely accurate mathematically. Be analyzing measured data of radial displacements as function of gantry angle, the formula calculates the MI position in room coordinate.« less

Preparation of course materials: Elementary mathematics of powered flight

NASA Technical Reports Server (NTRS)

Rublein, George T.

1994-01-01

Non-science students at William and Mary will soon be required to take a mathematics course in order to earn a bachelor's degree. A standard menu of technique courses is the usual way in which universities provide for this requirement: Trigonometry, probability, geometry for teachers, and the like. In this work, we attempt to break away from these largely unsuccessful choices. Our intent is to prepare material that sets a variety of simple mathematical procedures in the context of a commonly experienced part of students' lives: riding in commercial airplanes. The work, begun last summer at Langley, is now close to completion and trial in upcoming fall term at William and Mary. As of this writing, the narrative is complete for 12 to 14 projected sections. We have prepared material on wind triangles, wind roses, navigation maps, drag induced loss of velocity for unpowered missiles (tennis balls), luggage and its effect on center of gravity, localized magnetic declination and VOR orientation, geometry of great circles, terminal velocity for falling bodies, pressure vessels: tires and balloons and blimps, global structure of declination lines, map projections (mercator, azimuthal equidistant, Lambert), ears and their reaction to altitude change. The next section will treat lift, drag and thrust. The last will treat control surfaces. The entire approach avoids any effort to investigate mathematical topics that arise in the solution of problems. And by the same token, we avoid any organized attempt to explain aeronautical engineering, even on an elementary level. We look only at enough mathematics to do a problem and we select only engineering topics that permit some kind of (elementary) mathematical analysis. In the end, we will think of the material as successful if two things happen: Students must come away with some confidence that even lay people can quantify parts of their surroundings. Other potential instructors must be willing to gain enough familiarity with the physical content of the material so that it can be used at other universities.

Biomechanical comparison of four C1 to C2 rigid fixative techniques: anterior transarticular, posterior transarticular, C1 to C2 pedicle, and C1 to C2 intralaminar screws.

PubMed

Lapsiwala, Samir B; Anderson, Paul A; Oza, Ashish; Resnick, Daniel K

2006-03-01

We performed a biomechanical comparison of several C1 to C2 fixation techniques including crossed laminar (intralaminar) screw fixation, anterior C1 to C2 transarticular screw fixation, C1 to 2 pedicle screw fixation, and posterior C1 to C2 transarticular screw fixation. Eight cadaveric cervical spines were tested intact and after dens fracture. Four different C1 to C2 screw fixation techniques were tested. Posterior transarticular and pedicle screw constructs were tested twice, once with supplemental sublaminar cables and once without cables. The specimens were tested in three modes of loading: flexion-extension, lateral bending, and axial rotation. All tests were performed in load and torque control. Pure bending moments of 2 nm were applied in flexion-extension and lateral bending, whereas a 1 nm moment was applied in axial rotation. Linear displacements were recorded from extensometers rigidly affixed to the C1 and C2 vertebrae. Linear displacements were reduced to angular displacements using trigonometry. Adding cable fixation results in a stiffer construct for posterior transarticular screws. The addition of cables did not affect the stiffness of C1 to C2 pedicle screw constructs. There were no significant differences in stiffness between anterior and posterior transarticular screw techniques, unless cable fixation was added to the posterior construct. All three posterior screw constructs with supplemental cable fixation provide equal stiffness with regard to flexion-extension and axial rotation. C1 lateral mass-C2 intralaminar screw fixation restored resistance to lateral bending but not to the same degree as the other screw fixation techniques. All four screw fixation techniques limit motion at the C1 to 2 articulation. The addition of cable fixation improves resistance to flexion and extension for posterior transarticular screw fixation.

"The Math You Need" When Faculty Need It: Enhancing Quantitative Skills at a Broad Spectrum of Higher Education Institutions

NASA Astrophysics Data System (ADS)

Baer, E. M.; Wenner, J. M.

2014-12-01

Implementation of "The Math You Need, When You Need It" (TMYN) modules at a wide variety of institutions suggests a broad need for faculty support in helping students develop quantitative skills necessary in introductory geoscience courses. Designed to support students in applying geoscience relevant quantitative skills, TMYN modules are web-based, self-paced and commonly assigned outside of class. They include topics such as calculating slope, rearranging equations, and unit conversions and provide several applications of the mathematical technique to geoscience problems. Each instructor chooses modules that are applicable to the content in his/her individual course and students typically work through the module immediately before the module topic is applied in lab or class. Instructors assigned TMYN modules in their courses at more than 40 diverse institutions, including four-year colleges and universities (4YCs) that vary from non-selective to highly selective and open-door two-year colleges (2YCs). Analysis of module topics assigned, frequency of module use, and institutional characteristics reveals similarities and differences among faculty perception of required quantitative skills and incoming student ability at variably selective institutions. Results indicate that institutional type and selectivity are not correlated with module topic; that is, faculty apply similar quantitative skills in all introductory geoscience courses. For example, nearly every instructor assigned the unit conversions module, whereas very few required the trigonometry module. However, differences in number of assigned modules and faculty expectations are observed between 2YCs and 4YCs (no matter the selectivity). Two-year college faculty typically assign a higher number of modules per course and faculty at 4YCs more often combine portions of multiple modules or cover multiple mathematical concepts in a single assignment. These observations suggest that quantitative skills required for introductory geoscience courses are similar among all higher-education institution types. However, faculty at 4YCs may expect students to acquire and apply multiple quantitative skills in the same class/lab, whereas 2YC faculty may structure assignments to introduce and apply only one quantitative technique at a time.

Development of a measure of student self-evaluation of physics exam performance

NASA Astrophysics Data System (ADS)

Hagedorn, Eric Anthony

The central purpose of this study was to provide preliminary evidence of the reliability and validity of the SEVSI - P (Self- evaluation scaled instrument - physics). This instrument, designed to measure student self-evaluation of physics exam performance, was developed in congruence with social cognitive theory. Self-evaluation in this study is defined to consist of two of the three subprocesses of self-regulation: self-observation and judgmental process. As such, the SEVSI - P consists of two subscales, one measuring the frequency and types of self-observations made during a physics exam and one measuring the frequency and types of judgmental comparisons made after an exam. Data from 621 completed surveys, voluntarily taken by first semester algebra/trigonometry based physics students at six Midwestern universities and one Southern university, were analyzed for reliability and factorial validity. Cronbach alphas of .71 and .83 for the self-observation and judgment subscales, respectively, indicate acceptable reliability for the instrument. Confirmatory factor analysis indicates the acceptability of the hypothesis that the data analyzed could have indeed been obtained from the proposed two factor model (self-observation and judgment). The results of this confirmatory factor analysis provide preliminary construct validity for this instrument. A number of theoretically related items were included on the SEVSI - P form to elicity information about the use of goals and pre-planned strategies, actions taken in response to previous poor performances, and emotional responses to performance. A correlational analysis of these items along with the self-observation and judgment subscale scores provided a limited degree of convergent validity for the two subscales. Analyses of variance were done to determine the presence of differences in scoring patterns based on gender or reported ethnic origin. These results indicate slightly higher judgment subscale scores for women and members of minority groups. The implications of these differences are suggested as warranting future research. Future uses of the SEVSI - P include classroom use to assist students self-evaluate their exam performances in order to increase their achievement. Future research using the SEVSI - P to determine the causal relationships between self-evaluation, actual achievement, and other social cognitive constructs such as self-efficacy are suggested.

Super-Resolution Enhancement From Multiple Overlapping Images: A Fractional Area Technique

NASA Astrophysics Data System (ADS)

Michaels, Joshua A.

With the availability of large quantities of relatively low-resolution data from several decades of space borne imaging, methods of creating an accurate, higher-resolution image from the multiple lower-resolution images (i.e. super-resolution), have been developed almost since such imagery has been around. The fractional-area super-resolution technique developed in this thesis has never before been documented. Satellite orbits, like Landsat, have a quantifiable variation, which means each image is not centered on the exact same spot more than once and the overlapping information from these multiple images may be used for super-resolution enhancement. By splitting a single initial pixel into many smaller, desired pixels, a relationship can be created between them using the ratio of the area within the initial pixel. The ideal goal for this technique is to obtain smaller pixels with exact values and no error, yielding a better potential result than those methods that yield interpolated pixel values with consequential loss of spatial resolution. A Fortran 95 program was developed to perform all calculations associated with the fractional-area super-resolution technique. The fractional areas are calculated using traditional trigonometry and coordinate geometry and Linear Algebra Package (LAPACK; Anderson et al., 1999) is used to solve for the higher-resolution pixel values. In order to demonstrate proof-of-concept, a synthetic dataset was created using the intrinsic Fortran random number generator and Adobe Illustrator CS4 (for geometry). To test the real-life application, digital pictures from a Sony DSC-S600 digital point-and-shoot camera with a tripod were taken of a large US geological map under fluorescent lighting. While the fractional-area super-resolution technique works in perfect synthetic conditions, it did not successfully produce a reasonable or consistent solution in the digital photograph enhancement test. The prohibitive amount of processing time (up to 60 days for a relatively small enhancement area) severely limits the practical usefulness of fraction-area super-resolution. Fractional-area super-resolution is very sensitive to relative input image co-registration, which must be accurate to a sub-pixel degree. However, use of this technique, if input conditions permit, could be applied as a "pinpoint" super-resolution technique. Such an application could be possible by only applying it to only very small areas with very good input image co-registration.

Aortoiliac morphologic correlations in aneurysms undergoing endovascular repair.

PubMed

Ouriel, Kenneth; Tanquilut, Eugene; Greenberg, Roy K; Walker, Esteban

2003-08-01

The feasibility of endovascular aneurysm repair depends on morphologic characteristics of the aortoiliac segment. Knowledge of such characteristics is relevant to safe deployment of a particular device in a single patient and to development of new devices for use in patients with a broader spectrum of anatomic variations. We evaluated findings on computed tomography scans for 277 patients being considered for endovascular aneurysm repair. Aortic neck length and angulation estimates were generated with three-dimensional trigonometry. Specific centerline points were recorded, corresponding to the aorta at the celiac axis, lowest renal artery, cranial aspect of the aneurysm sac, aortic terminus, right hypogastric artery origin, and left hypogastric origin. Aortic neck thrombus and calcium content were recorded, and neck conicity was calculated in degrees. Statistical analysis was performed with the Spearman rank correlation. Data are expressed as median and interquartile range. Median diameter of the aneurysms was 52 mm (interquartile range, 48-59 mm) in minor axis and 56 mm (interquartile range, 51-64 mm) in major axis, and median length was 88 mm (interquartile range, 74-103 mm). Median proximal aortic neck diameter was 26 mm (interquartile range, 22-29 mm), and median neck length was 30 mm (interquartile range, 18-45 mm). The common iliac arteries were similar in diameter (right artery, 16 mm [interquartile range, 13-20 mm]; left artery, 15 mm [interquartile range, 11-18 mm]) and length (right, 59 mm [interquartile range, 50-69 mm]; left, 60 mm [interquartile range, 49-70 mm]). Median angulation of the infrarenal aortic neck was 40 degrees (interquartile range, 29-51 degrees), and median angulation of the suprarenal segment was 45 degrees (interquartile range, 36-57 degrees). By gender, sac diameter, proximal neck diameter, and iliac artery diameter were significantly larger in men. Significant linear associations were identified between sac diameter and sac length, neck angulation, and iliac artery diameter. As the length of the aneurysm sac increased the proximal aortic neck length decreased. Conversely, as the sac length decreased sac eccentricity increased. Mural thrombus content within the neck increased with increasing neck diameter. There is considerable variability in aortoiliac morphologic parameters. Significant associations were found between various morphologic variables, links that are presumably related to a shared pathogenesis for aberration in aortoiliac diameter, length, and angulation. Ultimately this information can be used to develop new endovascular devices with broader applicability and improved long-term results.

The use of trigonometry in bloodstain analysis.

PubMed

Makovický, Peter; Horáková, Petra; Slavík, Petr; Mošna, František; Pokorná, Olga

2013-04-01

Bloodstain pattern analysis (BPA) is a valid forensic method which belongs to the category of biological methods using trigonomic models. Despite its development through the years, the method has been re-formulated a standard one and globally used, recognized in standard sheets. This method permits exact analysis of the dynamic and characteristic properties of bloodstains after impact on surfaces such as floors, walls, and ceilings, various exterior and interior items, and clothes. It is also possible to determine the characteristics of blood from the outer part of the body. According to the presence of blood and its quantity, it is also possible to use this method for verification of reconstruction of criminal acts, while being tested for its validity with primary conditions of preserved and readable traces of blood. Even though this method is not considered as the major one or the only one information obtained in this way can be used for judicial. In our research, we tested the validity of this method in an experimental model using firearms. We compared measurements of the lengths of trajectory of impact and the height of the blood sprayed upwards from a distance of 1, 3, 5 and 10 meters. The experiment was based on two main presumptions. The first was the knowledge of the value of the distance and the angle of impact of the bloodstain, the second, the ability of the blood to reach a certain height and the angle of its impact. In accordance with trigonometric formulas, both the impact of the selected distance of drops of blood, and the height of the selected bloodstain could be determined without any verification of the flight trajectory and the distance of bloodstains. The results indicate that the method for these requirements differs from the real values, while increasing the measurements with the indicated spot of the shot. Aside from the unique values which were calculated, other results of the impact of the distance of drops of bloodstain were considered of lower value, and the values concerning the height of the bloods stains after the shot higher than real values. In spite of the lack of total accuracy, we recommend using this method widely and more often for investigation and verification of individual acts in criminal and forensic practice.

Technological pedagogical content knowledge and teaching practice of mathematics trainee teachers

NASA Astrophysics Data System (ADS)

Tajudin, Nor'ain Mohd.; Kadir, Noor Zarinawaty Abd.

2014-07-01

This study aims to identify the level of technological pedagogical content knowledge (TPCK) of mathematics trainee teachers at Universiti Pendidikan Sultan Idris (UPSI) and explore their teaching practices during practical training at school. The study was conducted in two phases using a mix-method research. In the first phase, a survey method using a questionnaire was carried out on 156 trainee teachers of Bachelor of Mathematics Education (AT14) and Bachelor of Science (Mathematics) with Education (AT48). The instrument used was a questionnaire that measures the level of content knowledge, pedagogy, technology and TPCK of mathematics. Data were analyzed using descriptive statistics, namely the mean. While in the second phase, the interview method involved four trainee teachers were performed. The instrument used was the semi-structured interview protocol to assess the trainee teacher's TPCK integration in their teaching practice. Data were analyzed using the content analysis. The findings showed that the level of knowledge of TPCK among trainee teachers was moderate with overall mean score of 3.60. This level did not show significant differences between the two programs with mean scores of 3.601 for the AT14 group and 3.603 for the AT48 group. However, there was a difference for gender classification such that the female trainees had mean score of 3.58 and male trainees with mean score of 3.72. Although students' TPCK level was moderate, the level of content knowledge (CK), technological knowledge (TK) and pedagogical knowledge (PK), showed a higher level with overall mean scores of 3.75, 3.87 and 3.84 respectively. The findings also showed that in terms of content knowledge, trainee teacher's learning mathematics background was good, but the knowledge of mathematics was limited in the curriculum, philosophy and application aspect. In terms of pedagogical content knowledge, all respondents tend to use lecture and discussion methods in teaching Trigonometry topic in almost the entire teaching and learning sessions. In terms of using technology in teaching and learning process, all respondents were more likely to use PowerPoint presentation as well as assisted by the use of a scientific calculator. The use of PowerPoint was limited in explaining the title and its content and the calculators were used during practice sessions. The four trainee teachers were still at the recognizing and acceptance levels of TPCK integration during their teaching practice. This study indicates that UPSI trainee teachers should be given exposure in the use of information and communication technology and trained to provide alternative teaching strategies in more creative ways if there are obstacles when integrating technology in teaching and learning process.

Documentation and evaluation of slope instabilities and other geological phenomena in the Geopark Bohemian Paradise (Czech Republic)

NASA Astrophysics Data System (ADS)

Krejčí, Oldřich; Krejčí, Vladimíra; Švábenická, Lilian; Hartvich, Filip

2016-04-01

Geographically, the area is part of the Bohemian Cretaceous Basin, the unit Jičín Hilly land. Since October 2005, the area belongs to the European Geopark UNESCO Bohemian Paradise. The reason of the protection is a major complex of rocks, natural forest communities and geomorphological valuable territory. The territory has been newly geologically mapped in a scale of 1 : 25,000. Sediments of the Czech Cretaceous Basin covers an area of 181 km2 and were deposited transgressively on the Permian - Carboniferous and crystalline basement of the Bohemian Massif. Except for locally developed basal sediments of fluvial origin they are mostly shallow marine sediments. Middle Turonian to Lower Coniacian rocks of the Jizera lithofacies are dominant by calcareous sandstones deposited under extremely dynamic conditions. Scattered alkaline volcanics penetrate the older formations as small intrusions and form locally preserved bodies at the surface. Area is strongly predisposed to the development of various types of landforms by structural segmentation of the Cretaceous sandstones and claystones and by Plio-Pleistocene inverse erosion. Numerous archival manuscripts are available from this area together with published geological, engineering-geological, geomorphological and historical papers. This is due to the fact that in 1926 a large landslide destroyed a substantial part of the village Dneboh, situated on the slope below a rock castle Drabske Svetnicky. Drabske Svetnicky is a ruin of a 13th century castle. It is located on the ragged edge of a sandstone cliff high above surrounding landscape. The castle covers a group of seven sandstone rocks, connected with wooden bridges. In the 50ies of the 20th century, an increased attention was paid to Drabske Svetnicky by experts on medieval architecture and a restoration of the original state of the castle rock was accomplished. Remnants of pottery and other findings suggest that the plateau region of the castle was first inhabited from the Neolithic period to the early Bronze Age (3000 to 1700 BC) almost continuously until the mid-15th century. Precise topographic maps of the rugged sandstone terrain and steep rock slopes do not exist. Our detailed 3D block model is based on the results of detailed geodetic survey of the situation within the rock castle and on laser trigonometry data. The area was further explored by air methods LIDAR (Light Detection And Ranging). Results of the LIDAR photos interpretation was used to clarify the extent of landslides in the environment of the rock city. In 1990, regular measurement of relative movements broken sandstone towers started in order to confirm or refute the current activity of slope movements. The acquired LIDAR data confirmed the continuous slope movements at an average rate of up to 2 mm/ year.

Measurement of one-way velocity of light and light-year

NASA Astrophysics Data System (ADS)

Chen, Shao-Guang

For space science and astronomy the fundamentality of one-way velocity of light (OWVL) is selfevident. The measurement of OWVL (distance / interval) and the clock synchronization with light-signal transfer make a logical circulation. This means that OWVL could not be directly measured but only come indirectly from astronomical method (Romer's Io eclipse and Bradley's sidereal aberration). Furthermore, the light-year by definitional OWVL and the trigonometry distance with AU are also un-measurable. In this report two methods of clock synchronization to solve this problem were proposed: The arriving-time difference of longitudinal-transverse wave (Ts - Tp) or ordinary-extraordinary light (Te - To) is measured by single clock at one end of a dual-speed transmission-line, the signal transmission-delay (from sending-end time Tx to receiving-end time Tp or To) calculated with wave-speed ratio is: (Tp -Tx) = (Ts -Tp) / ((Vp / Vs) - 1) or: (To -Tx) = (Te - To) / ((Vo / Ve ) - 1), where (Vp / Vs) = (E / k) 1/2 is Yang's / shear elastic-modulus ratio obtained by comparing two strains at same stress, (Vo / Ve) = (ne / no) is extraordinary/ordinary light refractive-index ratio obtained by comparing two deflection-angles. Then, two clocks at transmission-line two ends can be synchronized directly to measure the one-way velocity of light and light-year, which work as one earthquakestation with single clock measures first-shake-time and the distance to epicenter. The readings Na and Nb of two counters Ca and Cb with distance L are transferred into a computer C by two leads with transmission-delay Tac and Tbc respectively. The computer progressing subtraction operation exports steady value: (Nb - Na) = f (Ta - Tb ) + f (Tac - Tbc ), where f is the frequency of light-wave always passing Ca and Cb, Ta and Tb are the count-start time of Ca and Cb respectively. From the transmission-delay possess the spatial translational and rotational invariability, the computer exports steady value: (Nb - Na)' = f (Ta - Tb ) - f ( Tac - Tbc) when two leads had been interchanged. Or: 2 f (Ta - Tb ) = (Nb - Na) + (Nb - Na)'. After Ca and Cb are successively closed by a count-stop pulse modulated into the light-wave, the immovable reading difference of two counters is f (Ta - Tb ) + N, N is the wave-number in length L. After delay Tac or Tbc the computer exports last steady value: (Nb - Na)" = f (Ta - Tb ) + N, Or : 2N = 2 (Nb - Na)" - (Nb - Na) - (Nb - Na)' . L / N is one-way wavelength l from Ca to Cb, simultaneously measuring the frequency f , l f is one-way velocity of light c + from Ca to Cb. To reverse the transmitting-direction of light the measuring result of l f is just one-way velocity of light c - from Cb to Ca. Leastways for 86 Kr light-wave c + = c - =c is valid. With classical Newtonian mechanics and ether wave optics, the one-way velocity of light can be measured in the Galileo coordinate system with isotropic length unit ——1889 international meter definition. Special relativity can entirely establish on the measuring results.

Implications of the field sampling procedure of the LUCAS Topsoil Survey for uncertainty in soil organic carbon concentrations.

NASA Astrophysics Data System (ADS)

Lark, R. M.; Rawlins, B. G.; Lark, T. A.

2014-05-01

The LUCAS Topsoil survey is a pan-European Union initiative in which soil data were collected according to standard protocols from 19 967 sites. Any inference about soil variables is subject to uncertainty due to different sources of variability in the data. In this study we examine the likely magnitude of uncertainty due to the field-sampling protocol. The published sampling protocol (LUCAS, 2009) describes a procedure to form a composite soil sample from aliquots collected to a depth of between approximately 15-20. A v-shaped hole to the target depth is cut with a spade, then a slice is cut from one of the exposed surfaces. This methodology gives rather less control of the sampling depth than protocols used in other soil and geochemical surveys, this may be a substantial source of variation in uncultivated soils with strong contrasts between an organic-rich A-horizon and an underlying B-horizon. We extracted all representative profile descriptions from soil series recorded in the memoir of the 1:250 000-scale map of Northern England (Soil Survey of England and Wales, 1984) where the base of the A-horizon is less than 20 cm below the surface. The Soil Associations in which these 14 series are significant members cover approximately 17% of the area of Northern England, and are expected to be the mineral soils with the largest organic content. Soil Organic Carbon content and bulk density were extracted for the A- and B-horizons, along with the thickness of the horizons. Recorded bulk density, or prediction by a pedotransfer function, were also recorded. For any proposed angle of the v-shaped hole, the proportions of A- and B-horizon in the resulting sample may be computed by trigonometry. From the bulk density and SOC concentration of the horizons, the SOC concentration of the sample can be computed. For each Soil Series we drew 1000 random samples from a trapezoidal distribution of angles, with uniform density over the range corresponding to depths 15-20 cm and zero density for angles corresponding to depths larger than 21 cm or less than 14 cm. We computed the corresponding variance of sample SOC contents. We found that the variance in SOC determinations attributable to variation in sample depth for these uncultivated soils was of the same order of magnitude as the estimate of the subsampling + analytical variance component (both on a log scale) that we previously computed for soils in the UK (Rawlins et al., 2009). It seems unnecessary to accept this source of uncertainty, given the effort undertaken to reduce the analytical variation which is no larger (and often smaller) than this variation due to the field protocol. If pan-European soil monitoring is to be based on the LUCAS Topsoil survey, as suggested by an initial report, uncertainty could be reduced if the sampling depth was specified to a unique depth, rather than the current depth range. LUCAS. 2009. Instructions for Surveyors. Technical reference document C-1: General implementation, Land Cover and Use, Water management, Soil, Transect, Photos. European Commission, Eurostat. Rawlins, B.G., Scheib, A.J., Lark, R.M. & Lister, T.R. 2009. Sampling and analytical plus subsampling variance components for five soil indicators observed at regional scale. European Journal of Soil Science 60, 740-747

BOOK REVIEW An Introduction to Relativity An Introduction to Relativity

NASA Astrophysics Data System (ADS)

Dray, Tevian

2011-02-01

As is clearly stated in the preface, this book is a fresh rewrite of Narlikar's published lecture notes [1] from 1978, resulting in a unique historical perspective with a modernized presentation. For workers in the field, this book provides a fascinating glimpse of the state of the art 30 years ago, but there are some inevitable idiosyncracies which might make it less useful to the beginner, especially those attempting to read this book on their own. The author is well-known for his informal, readable style, and this book is no exception. Although the treatment is overly algebraic from this reviewer's modern perspective, that is not necessarily a drawback, and does accurately reflect contemporary usage. This text is a `math first' approach to relativity, presenting the tensor calculus first, and the physics second. Not surprisingly given the author's interests, there is an emphasis on cosmological applications. After a quick review of special relativity, the book jumps right in with a fairly standard introduction to tensor calculus, followed almost immediately by chapters on Einstein's equations, the Schwarzschild solution, experimental tests, relativistic astrophysics, and black holes, followed in turn by several chapters on cosmology. This new text is a faithful adaptation of the earlier text, with essentially identical tables of contents. In fact, the only notable differences are the opening chapter summarizing special relativity, a brief new chapter on gravitational radiation, and the exclusion of a lecture on the Steady State Theory — and that material appears separately in the author's cosmology text [2]. Some of the material, such as the excellent chapter on symmetries, is largely identical to the original, but most of the material has been extensively rewritten, with most of the changes being to presentation, rather than content. There are some seldom seen gems in this book, such as the aforementioned discussion of symmetries, the extensive discussion of cosmology, a formula for Lorentz transformations in an arbitrary direction, and numerous discussions of early work in the field, complete with original references. My only real quibble with this book (apart from the use of signature (+ - - -) is that, without comment, it makes some rather sophisticated assumptions about the reader. Chapter 1, the self-described `crash course' in special relativity, expects the reader to be comfortable with standard but nontrivial manipulations of Maxwell's equations, with hyperbolic trigonometry, and to be able to comprehend a statement that Newton's first law `remains as it is' in special relativity, without any statement of the law itself. Tensor notation is also introduced much too quickly in this chapter, although this topic is revisited later in more detail. These are not unreasonable assumptions about the intended audience, who are likely to be graduate students in physics or workers in the field. But the addition of a few explanatory remarks, or an explicit statement of the intended audience, might have encouraged the less experienced reader to keep reading. In summary, this is an excellent book, which researchers in relativity will treasure for its historical perspective. Instructors of a first course in general relativity who are seeking a `math first' approach should definitely take a look, although they should be prepared to explain some missing details in the presentation. References [1] Narlikar J V 1979 Lectures on General Relativity and Cosmology (London: Macmillan) [2] Narlikar J V 2002 An Introduction to Cosmology (Cambridge: Cambridge University Press)

Direct measurement of Lorentz transformation with Doppler effects

NASA Astrophysics Data System (ADS)

Chen, Shao-Guang

For space science and astronomy the fundamentality of one-way velocity of light (OWVL) is selfevident. The measurement of OWVL (distance/interval) and the clock synchronization with light-signal transfer make a logical circulation. This means that OWVL could not be directly measured but only come indirectly from astronomical method (Romer's Io eclipse and Bradley's sidereal aberration), furthermore, the light-year by definitional OWVL and the trigonometry distance with AU are also un-measurable. For to solve this problem two methods of clock synchronization were proposed: The direct method is that at one end of dual-speed transmissionline with single clock measure the arriving-time difference of longitudinal wave and transverse wave or ordinary light and extraordinary light, again to calculate the collective sending-time of two wave with Yang's /shear elastic-modulus ratio (E/k) or extraordinary/ordinary light refractive-index ratio (ne/no), which work as one earthquake-station with single clock measures first-shake time and the distance to epicenter; The indirect method is that the one-way wavelength l is measured by dual-counters Ca and Cb and computer's real-time operation of reading difference (Nb - Na) of two counters, the frequency f is also simultaneously measured, then l f is just OWVL. Therefore, with classical Newtonian mechanics and ether wave optics, OWVL can be measured in the Galileo coordinate system with an isotropic length unit (1889 international meter definition). Without any hypotheses special relativity can entirely establish on the metrical results. When a certain wavelength l is defined as length unit, foregoing measurement of one-way wavelength l will become as the measurement of rod's length. Let a rigidity-rod connecting Ca and Cb moves relative to lamp-house with velocity v, rod's length L = (Nb - Na) l will change follow v by known Doppler effect, i.e., L(q) =L0 (1+ (v/c) cos q), where L0 is the proper length when v= 0, v• r = v cos q, r is the unit vector from lamphouse point to counters. Or: L (0) L (pi) =L0 (1+(v/c)) L0 (1 - (v/c)) =L0 2 y2 =L2 Or: L ≡ [L(0)L(pi)]1/2 =L0 y , which y ≡ (1 - (v/c)2 )1/2 is just Fitzgerald-Lorentzian contraction-factor. Also, when a light-wave period p is defined as time unit, from Doppler's frequency-shift the count N with p of one period T of moving-clock is: T(q) = N(q) p = T0 /(1+(v/c) cos q) Or: T ≡ (T(0) T(pi))1/2 = T 0 /y , where T0 is the proper period when v = 0, which is just the moving-clock-slower effect. Let r from clock point to lamp-house ((v/c) symbol reverse), Doppler formula in the usual form is: f (q) = 1/T(q) = f0 (1 - (v/c) cos q). Therefore, Lorentz transformation is the square root average of positive and negative directions twice metrical results of Doppler's frequency-shift, which Doppler's once items ( positive and negative v/c ) are counteract only residual twice item (v/c)2 (relativity-factor). Then Lorentz transformation can be directly measured by Doppler's frequency-shift method. The half-life of moving mu-meson is statistical average of many particles, the usual explanation using relativity-factor y is correct. An airship moving simultaneously along contrary directions is impossible, which makes that the relativity-factor y and the twin-paradox are inexistent in the macroscopical movement. Thereby, in the navigations of airship or satellite only use the measurement of Doppler's frequency-shift but have no use for Lorentz transformation.

Radio Astronomers Set New Standard for Accurate Cosmic Distance Measurement

NASA Astrophysics Data System (ADS)

1999-06-01

A team of radio astronomers has used the National Science Foundation's Very Long Baseline Array (VLBA) to make the most accurate measurement ever made of the distance to a faraway galaxy. Their direct measurement calls into question the precision of distance determinations made by other techniques, including those announced last week by a team using the Hubble Space Telescope. The radio astronomers measured a distance of 23.5 million light-years to a galaxy called NGC 4258 in Ursa Major. "Ours is a direct measurement, using geometry, and is independent of all other methods of determining cosmic distances," said Jim Herrnstein, of the National Radio Astronomy Observatory (NRAO) in Socorro, NM. The team says their measurement is accurate to within less than a million light-years, or four percent. The galaxy is also known as Messier 106 and is visible with amateur telescopes. Herrnstein, along with James Moran and Lincoln Greenhill of the Harvard- Smithsonian Center for Astrophysics; Phillip Diamond, of the Merlin radio telescope facility at Jodrell Bank and the University of Manchester in England; Makato Inoue and Naomasa Nakai of Japan's Nobeyama Radio Observatory; Mikato Miyoshi of Japan's National Astronomical Observatory; Christian Henkel of Germany's Max Planck Institute for Radio Astronomy; and Adam Riess of the University of California at Berkeley, announced their findings at the American Astronomical Society's meeting in Chicago. "This is an incredible achievement to measure the distance to another galaxy with this precision," said Miller Goss, NRAO's Director of VLA/VLBA Operations. "This is the first time such a great distance has been measured this accurately. It took painstaking work on the part of the observing team, and it took a radio telescope the size of the Earth -- the VLBA -- to make it possible," Goss said. "Astronomers have sought to determine the Hubble Constant, the rate of expansion of the universe, for decades. This will in turn lead to an estimate of the age of the universe. In order to do this, you need an unambiguous, absolute distance to another galaxy. We are pleased that the NSF's VLBA has for the first time determined such a distance, and thus provided the calibration standard astronomers have always sought in their quest for accurate distances beyond the Milky Way," said Morris Aizenman, Executive Officer of the National Science Foundation's (NSF) Division of Astronomical Sciences. "For astronomers, this measurement is the golden meter stick in the glass case," Aizenman added. The international team of astronomers used the VLBA to measure directly the motion of gas orbiting what is generally agreed to be a supermassive black hole at the heart of NGC 4258. The orbiting gas forms a warped disk, nearly two light-years in diameter, surrounding the black hole. The gas in the disk includes water vapor, which, in parts of the disk, acts as a natural amplifier of microwave radio emission. The regions that amplify radio emission are called masers, and work in a manner similar to the way a laser amplifies light emission. Determining the distance to NGC 4258 required measuring motions of extremely small shifts in position of these masers as they rotate around the black hole. This is equivalent to measuring an angle one ten-thousandth the width of a human hair held at arm's length. "The VLBA is the only instrument in the world that could do this," said Moran. "This work is the culmination of a 20-year effort at the Harvard Smithsonian Center for Astrophysics to measure distances to cosmic masers," said Irwin Shapiro, Director of that institution. Collection of the data for the NGC 4258 project was begun in 1994 and was part of Herrnstein's Ph.D dissertation at Harvard University. Previous observations with the VLBA allowed the scientists to measure the speed at which the gas is orbiting the black hole, some 39 million times more massive than the Sun. They did this by observing the amount of change in the wavelength of the radio waves caused by the Doppler effect. The gas is orbiting at a speed of more than two million miles per hour. The orbiting disk of gas is almost edge-on as viewed from Earth. The astronomers obtained the orbital speeds and the positions of the masers in the disk by measuring the Doppler Shift of the masers at the disk's sides, where the gas is moving almost directly away from the Earth on one side and toward the Earth on the other. Measurements of the different orbital speeds at different distances from the black hole, made in 1994, allowed them to determine the mass of the black hole. These measurements required the great resolving power, or ability to see fine detail, of the VLBA. This picture of an orbiting disk was confirmed by measurement of centrifugal acceleration, according to the scientists. The newest observations were focused on maser "spots" on the near edge of the disk, where orbital motion shifts their position in the sky, though by an extremely small amount. The VLBA, however, was able to detect this extremely small movement, called "proper motion" by astronomers. This motion was detected by observing the galaxy at 4- to 8-month intervals over more than three years. "By knowing the speed at which the gas is orbiting and then measuring its motion across the sky, we can use plain old trigonometry to calculate the distance," Greenhill said. He added, however, that "you need a bit of luck to be able to do this. So far, we know of only 22 galaxies with water masers in their nuclear regions that also are relatively nearby. Then, the geometry of the disk, relative to Earth, has to be right to allow us to make such a measurement" The VLBA measurement of NGC 4258's distance differs significantly from the distance to that galaxy determined through HST observations of Cepheid variable stars. Using such stars, a team of astronomers led by University of California-Berkeley scientist Eyal Maoz has made preliminary and as-yet unpublished estimates of the distance to NGC 4258 as either 27 or 29 million light-years, depending on assumptions about the characteristics of this type of star in that galaxy. Other Cepheid-based galaxy distances were used to calculate the expansion rate of the universe, called the Hubble Constant, announced by a team of HST observers last week. "This difference could mean that there may be more uncertainty in Cepheid-determined distances than people have realized," said Moran. "Providing this directly-determined distance to one galaxy -- a distance that can serve as a milestone -- should be helpful in determining distances to other galaxies, and thus the Hubble Constant and the size and age of the universe" The VLBA is a system of ten radio-telescope antennas, each 25 meters (82 feet) in diameter, stretching some 5,000 miles from Mauna Kea in Hawaii to St. Croix in the U.S. Virgin Islands. Operated from NRAO's Array Operations Center in Socorro, NM, the VLBA offers astronomers the greatest resolving power of any telescope anywhere. The NRAO is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. Background information: Determining Cosmic Distances Determining cosmic distances obviously is vital to understanding the size of the universe. In turn, knowing the size of the universe is an important step in determining its age. "The size puts a limit on how much expansion could have occurred since the Big Bang, and thus tells us something about the age," said Moran. However, determining cosmic distances has proven to be a particularly thorny problem for astronomers. In the third century, B.C., the Greek astronomer Aristarchus devised a method of using trigonometry to determine the relative distances of the Moon and Sun, but in practice his method was difficult to use. Though a great first step, he missed the mark by a factor of 20. It wasn't until 1761 that trigonometric methods produced a relatively accurate distance to Venus, thus calibrating the size of the Solar System. The first accurate distance to another star was determined trigonometrically by Friedrich Wilhelm Bessel in 1838. Traditional trigonometric methods of measuring celestial distances require extremely accurate measurement of an object's position in the sky. By measuring the apparent shift in an object's position, called parallax, caused by the Earth's journey around the Sun, the distance to the object can be calculated. Until recent years, such measurements were limited by the atmosphere's degrading effect on optical observations. Recently, the Hipparcos satellite has measured stellar distances accurate to within 10 percent out to about 300 light-years. Beyond the range of parallax measurements, astronomers were forced to use indirect methods of estimating distances. Many of these methods make presumptions about the intrinsic brightness of objects, then estimate the distance by measuring how much fainter they appear on Earth. The faintness is presumed to be caused by the distance, according to the inverse-square law (doubling of the distance reduces brightness by a factor of four). Thus, stars of a particular spectral class are all presumed to be of the same intrinsic brightness. Such techniques have been used to estimate distances of stars out to about 25,000 light-years, still not far enough to estimate distance beyond our own Milky Way Galaxy. Early in the 20th Century, Henrietta Leavitt, of Harvard College Observatory, discovered that variable-brightness stars known as Cepheid variables showed a useful property -- the longer their pulsation periods, the brighter they are intrinsically. Once the absolute distance to a few Cepheids was determined, these stars were used to measure distances beyond the Milky Way. In the 1920s, Edwin Hubble used Cepheid-variable distance determinations to show that, contrary to then-prevalent opinion, many "nebulae" were, in fact, other galaxies far distant from our own. Distances determined using Cepheid variables, along with measurements of the Doppler shift of other galaxies' light, allowed Hubble to discover the expansion of the universe, the basis of the Big Bang theory. The Cepheid technique still is one of the building blocks of the extragalactic distance scale. However, because of absorption of light by interstellar dust and subtle differences among the stars themselves, this technique is subject to considerable uncertainty. Similarly, techniques that use a specific type of supernova (Type Ia) presumed to be of uniform intrinsic brightness, while able to make distance estimates farther than the Cepheid technique, still are subject to uncertainties. The NSF's VLBA, with resolving power hundreds of times better than even the Hubble Space Telescope, has allowed direct trigonometric techniques to be applied in measuring much greater distances than ever before. The VLBA measured the expansion of the shell of exploding debris from the supernova SN 1993J in the galaxy M81, 11 million light- years away. This information, combined with optical observations that measured the speed of the expanding debris by the Doppler shift of its emitted spectral lines, allowed a trigonometric calculation of the distance to M81. Now, with the VLBA's direct measurement of motions in the gas disk surrounding NGC 4258, trigonometric measurement, not subject to the vagaries of dust absorption and other uncertaintities in an object's brightness, has been extended to a distance of more than 23 million light-years.

Obituary: George Hamilton Bowen Jr. (1925-2009)

NASA Astrophysics Data System (ADS)

Willson, Lee Anne; Struck, Curtis

2011-12-01

Our colleague and collaborator George Hamilton Bowen, Jr., passed away November 1, 2009 in Ames, Iowa. George was born June 20, 1925 in Tulsa, Oklahoma to George and Dorothy (Huntington) Bowen. He married Marjorie Brown June 19, 1948 in Redondo Beach, California; they had five children, with eight grandchildren and five great-grandchildren at the time of his death. George H. Bowen's third or perhaps his fourth career was in astronomy. He was drafted into the navy in 1944, at the end of his first year as a student at Caltech, and ended his war-time service as an electronic technician on the aircraft carrier Shangri-La. He later said "In just nine months, starting from scratch (Ohm's law!), we learned an amazing amount - not by memorization, of course, but by study and real understanding of the basic function of the most advanced AC circuits then being used for instrumentation, measurements, communications, control systems, and much more." He gained a confidence that he could quickly and accurately diagnose and solve technical problems that stood him well in future work. One accomplishment he took particular pride in was figuring out how the radar control used cams and gears to solve the trigonometry for accurate pointing. He also described how the captain was alarmed when weather conditions changed so that refraction no longer showed them distant, small boats around the curvature of Earth. After the war, George Bowen returned to undergraduate and eventually graduate study at Caltech, where he was recruited to the biophysics research group headed by future Nobel Laureate Max Delbrück. George often described his joy in working with these first-rate scientists and finding himself accepted as a part of the effort. He finished his BS with honors in 1949 and his PhD in 1953 with a thesis on "Kinetic Studies on the Mechanism of Photoreactivation of Bacteriophase T2 Inactivated by Ultraviolet Light" involving work with E Coli. This work was supported by grants from the U. S. Atomic Energy Commission and the National Foundation for Infantile Paralysis. George began a postdoctoral appointment in 1952 at Oak Ridge National Laboratory, still in biophysics, before being recruited to the Department of Physics at Iowa State in 1954. While he became Emeritus Professor in 1993, he continued with his research until health considerations took primacy around 2007. When the biophysics effort in the Department of Physics split to form its own department in the early 1960s, George Bowen chose to stay with physics, commenting later that "It was clear to me that the important work was going to be in biochemistry and that was not my interest." At that time he and his wife, recovering from polio, were parenting a large family, and this may also have influenced his choice. He spent the next decade focusing on improving the teaching of physics at Iowa State, revising and documenting the lecture demonstrations, teaching a generation of physics and engineering majors (including some later very successful scientists), and giving himself time for family and hobbies, notably color photography with a laboratory at home. In the 1970s, George Bowen was ready to return to research, and found himself interested in the work being done in the astronomy program. Astronomy was a relatively recent addition to the department with just a few members. The newest arrival, Lee Anne Willson, was collaborating with Stephen Hill, an assistant professor at Michigan State and, coincidentally, one of the students who had benefited from Bowen's physics courses at Iowa State. Bowen became intrigued by the physical problem of the relation of pulsation and mass loss in pulsating red giants. When Hill left astronomy to become a geophysicist with the oil company Conoco, George Bowen volunteered to take over the code he had written. Bowen quickly mastered the science and the technical skills needed for this project. A number of factors probably made this possible: In addition to his native intelligence and insight, he was building on experience with complex systems in biology, a clarity derived from his teaching experience, and his problem-solving introduction in the Navy. After six months of working with Hill's code, Bowen started over and constructed his own code. What distinguished the Bowen code from other programs that have been written to similar purpose was his consistent use of simple but reasonable approximations to allow study of the complex interactions among physical processes that contribute to mass-loss. This made a nimble code well suited to determining the essential physics of the process that takes most stars from red giants to white dwarfs, and keeps stars in the range of 1.4-4 solar masses from exploding as supernovae. Although Bowen did not publish many papers in his relatively short astronomy career, his 1988 "Dynamical modeling of long- period variable star atmospheres" describing the physics is still a much-cited classic, and his 1991 paper with Willson, "From wind to superwind - The evolution of mass-loss rates for Mira models," laid out the essential properties of this important evolutionary stage. George Bowen contributed to astronomy also as the main organizer for the 1986 American Astronomical Society meeting in Ames, IA. He was always ready to help a student or a colleague, taking a lot of time over a referee report or response to a letter of inquiry. Those of us privileged to work closely with him feel his continued influence in our approach to problems and our interactions with our students. Devoted to his family, a man of high intellectual integrity and talent, George H. Bowen's legacy remains at Iowa State University and in the field of stellar astronomy.

Post-16 update

NASA Astrophysics Data System (ADS)

1999-09-01

Institute of Physics post-16 initiative, shaping the future Physics education based in IT - sorry ICT! Which, of course, makes for a nifty headline, but as is so often the case with neat slogans, little else. Two formative questions: What, in another decade, will the government of the day call `What you lot should be doing with computers'? and What, of the many things that currently exercise our intellects, will there be anything special to say about in a decade, concerning the use of computers in the teaching of physics? Advancing Physics represents some attempts to come to grips with the second of the two questions above. The first is left to a higher wisdom. In thinking about learning, what can we do with the huge processing power, increasingly available in smaller and less obtrusive packages? What will we do that helps people learn physics, both tomorrow and in 2009? Here are a few suggestions based on development work so far. Wide, reliable and shared access to well ordered learning resources. We have created a CD-ROM, with versions for both student and teacher, that provides a wide range of resources. These do not teach, but do provide. A commitment to allowing a course to evolve and adapt. Electronic publication puts the costs into origination, and not into publication and distribution. A website allows for the community of users to contribute. E-mail networks support individuals and propagate good practice. You can create and explore your own microworlds. The interactive nature of models, and the crafted relationships between the models and the natural world give an insight into the creative imaginary worlds of the physicist. The unreasonable, but pleasurable, success of mathematics in describing the natural world can come to the fore. Measurements that were not possible before are now possible. What was previously indirect, and inaccessible, now becomes a direct measurement, making relationships transparent in new and fruitful ways. The dichotomy between measured and fundamental variables can be resolved along a number of different axes. Shaping data to create understanding is also possible. Plotting, re-expressing and re-plotting can all happen within a time-span that retains the connections to the original physical situation. Making sense of the data can become a more active and integral part of doing physics. Two advantages for physics teachers: Firstly, the computer is another piece of equipment - one amongst the many that we already deploy based on our considerable experience. It may be a better intellectual prosthetic than some, but it is just one amongst many: good for some things, not to be let near others. Secondly, physics necessarily describes nature. Thus the chance of losing contact with our roots in a virtual canopy is slim; nurture and insight will always come from interaction with the physical. A summative question: Which, of the information and communication technologies available to students, is the most effective in stimulating, promoting and nurturing learning? (The more astute reader will notice the discontinuity between the implications of the embracing title, ICT, and the rather narrower, what we do with computers, somewhat closer to the current obsessions, rather meekly followed by the author in this piece. This final question has only one correct answer, several ramifications, and seeks to redress the balance lacking in the rest of this brief survey.) Ian Lawrence Maps and vectors Where would we be without vectors? Vectors are vital to a proper description of a variety of phenomena in physics. They are also notoriously difficult for students to come to terms with. Many students meet vectors for the first time in their A-level physics course when they learn mechanics. And this topic is often taught first. The vectors they meet - velocity, acceleration, force - can seem rather abstract, and acceleration in particular, being a rate of change of a rate of change, is fraught with conceptual difficulties. When students meet vectors, they are taught to add them, to subtract them to find components in various directions. This means they have to come to grips with Pythagoras's theorem and with trigonometry at the same time as learning the rules about how vectors behave: the magnitude of the sum is not the same as the sum of the magnitudes, for example. Rather than developing ideas about vectors at the same time as the difficult ideas encapsulated in Newton's laws, perhaps it might be better to develop a physical understanding of what vectors are and of how they combine first. One way of doing this is by considering maps: these represent, on a two-dimensional sheet of paper, the topology of a landscape. A road map represents a layout of towns, and by considering (straight line) journeys from town to town, ideas about vector addition can be developed: going from town A to town C via town B can be represented by the addition of two displacement vectors. And if these three towns are not in a straight line, then the magnitude of the sum is not the same as the sum of the magnitudes (the magnitude here is the straight line distance, of course). What's more, knowing the straight line distances between a small collection of towns - similar tables of distances can be found in many road atlases - allows the map to be reconstructed. But unless you know which way is north, the distances themselves give only the relative positions of the towns. Notice that these relative positions do not depend on the coordinate system (determined by our choice of north) so that vector sums are coordinate independent. Introducing a coordinate system - defined by compass directions - and thinking about how far north or east a particular journey takes you leads to ideas about components. And considering a map of mountainous country means that you can think about how three components can be represented on a flat piece of paper. This leads to generalizations of Pythagoras's theorem in three dimensions. Perhaps these ideas might help students to develop an intuitive understanding of what vectors are about. When they meet more complex and abstract ideas they might, then, see their way through the mathematical representations to the underlying physics. We might have to spend more time teaching the basics, but the pay-off could be a much better understanding later. Simon Carson

European team gauges a gamma-ray star

NASA Astrophysics Data System (ADS)

1996-03-01

Italian astrophysicists have pushed the Hubble Space Telescope to the limit of its powers in finding the distance of Geminga, a pointlike object 500 light-years from the Earth. It is the prototype of a novel kind of star, a radio-silent neutron star, which may be much more common in the Universe than previously supposed. Geminga is so weak in visible light that Hubble had to stare at the spot for more than an hour to register it adequately. The object is nevertheless one of the brightest sources of gamma-rays in the sky, and its output of this very energetic form of radiation can now be accurately ganged. Neutron stars, first discovered as radio pulsars in 1967, are fantastic creations of exploding stars, just one step short of a black hole. They are heavier than the Sun yet only about twenty kilometres wide. Made of compressed nuclear matter, they have gravity and magnetic fields many billions of times stronger than on the Earth. With the first direct measurement of the distance of a radio-silent neutron star, astrophysicists can assess Geminga's power and speed of motion. The astronomical task was like judging the width of a one- franc piece in Paris, seen from the distance of Sicily. Geminga's low brightness greatly aggravated the difficulties. Patrizia Caraveo and her colleagues at the Istituto di Fisica Cosmica in Milan arranged for Hubble's wide-field camera (WFPC2) to make its prolonged observations of Geminga three times. Their findings will be published in Astrophysical Journal Letters on 20 April 1996. Caraveo's co-authors are Giovanni Bignami and Roberto Mignani of Milan, and Laurence Taff of Johns Hopkins University, Maryland. The Italians took advantage of the European Space Agency's collaboration with NASA in the Hubble mission, which gives European astronomers privileged access to the Space Telescope. Shifts of millionths of a degree The three sightings of Geminga, made at intervals of six months, revealed small shifts in the position of the faint neutron star. The background of a few brighter but more distant stars displayed by Hubble's camera provided a frame of reference. In the course of a year, Geminga moved northeastwards by 0.17 arc-seconds, equivalent to one degree in 21,000 years. That was due to Geminga's high-speed motion through the Galaxy. But the neutron star also seemed to shift to either side of its interstellar track, because of changes in Hubble's viewpoint as the Earth orbited around the Sun. At opposite seasons, spring and autumn in this case, the Earth is on opposite sides of the Sun, at vantage points 300 million kilometres apart. As a result, the bearings of stars change a little, by the effect called parallax. Nearby stars shift more than very distant stars, and astronomers can measure their distances by trigonometry. The sideways displacement measured using the Hubble Space Telescope was 0.00636 arc-seconds, less than two millionths of a degree. From this figure the astronomers calculate that Geminga is 512 light-years away (157 parsecs) with an uncertainty of the order of 100 light-years. The strong gamma-rays and weak light observed today left Geminga around the time that Columbus discovered America. "We are pleased to have measured a parallax for an object at the limit of detection, which was never done before," says Patrizia Caraveo. "But what really matters is that we have pinned down an important object that has puzzled us for more than 20 years." From "it's not there" to "here it is" Geminga has tested the patience of the Milanese astronomers for twenty years. NASA's shortlived SAS-2 satellite (1973) first recorded a mysterious source of gamma-rays in the constellation of Gemini. In 1976 Giovanni Bignami named it Geminga. This is a pun signifying either "Gemini gamma" or "gh'è minga" which in the Milanese argot means "it's not there". Geminga lived up to its name when Bignami and his colleagues looked in vain for radio emissions from it. By 1981, data from ESA's longlived gamma-ray satellite COS-B had defined the position of the gamma-ray source to within half a degree -- well enough to prompt renewed efforts to identify it. Radio searches still drew a blank, but in the early 1980s Bignami and others found X-rays coming from Geminga in observations with NASA's Einstein satellite. They narrowed down Geminga's position to within a twentieth of a degree. There was no obvious counterpart to Geminga in visible light. Between 1983 and 1987 the Milanese team hunted for it with large telescopes in Hawaii and Chile. Eventually they selected a very faint object, peculiar in colour, as the visible Geminga. In 1992 a further sighting from Chile established Geminga's rate of movement across the sky. Meanwhile, the German/US/UK satellite Rosat revealed that Geminga pulsates in X-rays four times a second -- every 237 milliseconds to be precise. The same pulsation was found in gamma-rays by NASA's Gamma-Ray Observatory. Bignami and his colleagues then returned to the gamma-ray data from ESA's COS-B. They found the pulsation hidden there too and were able to compute the slowdown in Geminga's pulse-rate. From the slowdown they estimated the age of Geminga at 340,000 years. The distance measurement completes the gradual transformation of the enigmatic gamma-ray source into a well-characterized object. The Italian team calculates that Geminga is travelling at a speed of at least 120 kilometres per second. The neutron star's radiation in gamma-rays and X-rays is equivalent in energy to ten times the visible light of the Sun. More importantly, the way in which the neutron star distributes its energy output at different wavelengths is now known. "Neutron stars are radio sources for only a small fraction of their lives," says Giovanni Bignami. "So while we know 700 pulsars, there are probably millions of radio-silent neutron stars like Geminga. Thousands of them may be among X-ray sources already known but so far unidentified. I look forward to searching for new Gemingas with EPIC, our set of X-ray cameras in ESA's XMM satellite due for launch in 1999." The Hubble Space Telescope is a project of international cooperation between NASA and the European Space Agency (ESA). The Space Telescope Science Institute is operated by the Association of Universities for Research in Astronomy, Inc. (AURA), for NASA, under contract with the Goddard Space Flight Center, Greenbelt, MD. Patrizia A. Caraveo - Istituto di Fisica Cosmica del CNR, Via Bass 20133 - MILANO (Italy) Giovanni F. Bignami - Istituto di Fisica Cosmica del CNR, Via Bass - 20133-MILANO (Italy) Dipartimento di Ingegneria Industriale, Univ. of Cassino, CASSINO, Roberto Mignani - Istituto di Fisica Cosmica del CNR, Via Bassini, 20133-MILANO (Italy) Universita di Milano, Italy Laurence G. Taff - Department of Physics and Astronomy, JHU, BALTIMORE (MD Research Supported in part by General Observer Grant GO 5484.01-93

Value of Hipparcos Catalogue shown by planet assessments

NASA Astrophysics Data System (ADS)

1996-08-01

The first detailed findings from Hipparcos, recently published in Astronomy and Astrophysics Letters, confirm the existence of planets around other stars. Hipparcos astronomers plucked out their data on three stars suspected of possessing attendant planets. Their distances, measured far more accurately than ever before, enables the astronomers to rule out, in two cases, the possibility that the supposed planets might be small stars. The discovery of alien planets in the first astronomical step towards fashioning a proper science out of the speculations about life beyond the solar system. Hipparcos makes a decisive contribution by setting an upper limit to their masses. Astronomers at the Geneva Observatory caused a sensation last year when they reported slight motions in the star 51 Pegasi, due to a massive planet orbiting around it. With a ground-based telescope they detected small shifts in the wavelength of light as 51 Pegasi moved slowly under the influence of its invisible companion. This year, astronomers at San Francisco State University confirmed the discovery and have subsequently reported two similar cases, in the stars 47 Ursae Majoris and 70 Virginis. Uncertainties about the orientation of the planets' orbits and the distances of the stars left a wide margin of doubt about the masses of the candidate planets. Accurate rangefinding by Hipparcos puts the star 47 Ursae Majoris at a distance of 46 light-years. Calculations then set an upper limit on the mass of the companion at 7 to 22 times the mass of Jupiter, the Sun's largest planet. The Sun itself is a thousand times more massive that Jupiter, and theorists believe that the smallest true star would have a mass of 80 Jupiters. Below that mass, the object cannot burn hydrogen in the nuclear fashion, which is the most characteristic source of energy for stars. In the range between 17 and 80 Jupiter masses an object is called a brown dwarf. It can in theory derive a little energy by burning heavy hydrogen, or deuterium. Even the "worst-case" mass quoted here for the companion of 47 Ursae Majoris, 22 Jupiter masses, is only a maximum, not a measurement. So the companion is almost certainly a true planet with less than 17 times the mass of Jupiter. For the star 70 Virginis, the distance newly established by Hipparcos is 59 light-years. Even on the least favourable assumptions about its orbit, the companion cannot have more than 65 Jupiter masses. It could be brown dwarf rather than a planet, but not a true star. Much more ambiguous is the result for 51 Pegasi. Its distance is 50 light-years and theoretically the companion could have more than 500 Jupiter masses, or half the mass of the Sun. This is a peculiar case anyway, because the companion is very close to 51 Pegasi. Small planets of the size of the Earth might be more promising as abodes of life than the large planets detectable by present astronomical methods. Space scientists are now reviewing methods of detecting the presence of life on alien planets by detecting the infrared signature of ozone in a planet's atmosphere. Ozone is a by-product of oxygen gas, which in turn is supposed to be generated only by life similar to that on the Earth. Meanwhile the detection of planets of whatever size is a tour de force for astronomers, and by analogy with the Solar System one may suppose that large planets are often likely to be accompanied by smaller ones. "Hipparcos was not conceived to look for planets," comments Michael Perryman, ESA's project scientist for Hipparcos, "and this example of assistance to our fellow-astronomers involves a very small sample of our measurements. But it is a timely result when we are considering planet-hunting missions for the 21st Century. The possibilities include a super-Hipparcos that could detect directly the wobbles in nearby stars due to the presence of planets." Hipparcos Catalogue ready for use The result from Hipparcos on alien planets coincides with the completion of the Hipparcos Catalogue and the distribution of the data to collaborating scientists. The Catalogue lists the positions, distances, motions and brightnesses of 118,000 stars, within a self-consistent framework for the whole sky. The results include remarkable detail on the orbits of double stars and the changing light output of variable stars. Thanks to the unprecedented accuracy of space observations with a specially conceived satellite, the Hipparcos Catalogue gives positions of the stars to much better than a millionth of a degree. This is over one hundred times more accurate than the most careful fixes of stars from observatories on the ground. With the Catalogue's completion, the Hipparcos mission achieves its purpose of revolutionizing astrometry, the positional science that has underpinned mankind's studies of the Universe since the satellite's namesake Hipparchus the Greek surveyed the sky in the 2nd Century BC. Another pioneer of astrometry was Tycho the Dane (16th Century AD) and he is commemorated in a second product of the Hipparcos mission. The Tycho Catalogue, which is nearing completion, lists a far larger number of stars, slightly more than a million. Their positions will be less precise than in the Hipparcos Catalogue, but still far better than the ground-based results, and will include valuable data on the colours of stars. The first chance to cull discoveries from the Hipparcos data resides with the teams in nine European countries who have laboured to achieve this breakthrough in astronomy. For some team members the effort goes back thirty years, to the first proposal of an astrometric satellite in France in 1966. ESA approved the mission in 1980 and Hipparcos was launched in August 1989. Despite finding itself in the wrong orbit, Hipparcos operated successfully until March 1993. Three more years have gone into data analysis, using computers in a dozen institutes across Europe for the most elaborate calculations in the history of astronomy. Members of the teams will now have privileged use of the data for nearly a year, before the release of the catalogues to the world-wide astronomical community in April 1997. The new precision in star-fixing promises an extraordinary harvest of new results, on subjects ranging from asteroids to cosmology. A vivid picture of the stars in motion in our corner of the Milky Way Galaxy is one expected outcome. Hipparcos has more than doubled the number of known variable stars, and has discovered many thousands of new double or multiple star systems. Striding the light-years by parallax The study of stars with candidate planets is a dramatic example of Hipparcos's new determinations of the distances of stars by the parallax principle. Many other discoveries will flow from it. Parallax is an unfamiliar name for a familiar concept, akin to stereoscopic vision. People judge distances in nearby scenes from the difference in direction of the two eyes when focused on an object. Military rangefinders use the same principle, with more widely separated optics. Astronomers adapt the Earth's orbit to make a huge rangefinder. At opposite seasons, the Earth is on opposite sides of the Sun, at vantage points 300 million kilometres apart. As a result, the bearings of stars change a little. Nearby stars shift more than very distant stars, and astronomers can measure their distances by trigonometry. Until now, unavoidable inaccuracies in observing the directions of stars from the ground have meant that the distances of only the closest stars can be measured directly by parallax, out to about 100 light-years. Even at short ranges the margin of uncertainty is often wide. Long chains of inference and observation extend the distance scale to much farther objects, including galaxies millions or billions of light-years away. From the resulting estimates, astronomers try to calculate the age of the Universe and arrive at conclusions about its origin and evolution. The foundation for these reckonings has been decidedly shaky. By measuring seasonal changes of direction to an accuracy of better than a millionth of degree, the Hipparcos mission has taken a great stride across the light-years and transformed the art of determining astronomical distances. Hipparcos has pushed the practical range of direct and accurate distance measurement out to about 1000 light-years. That means rangefinding for a thousand times as many stars as before. As for the nearer stars, parallax measurements of their distances are now far more accurate, thanks to Hipparcos. The star 70 Virginis, which figures in the planets study, illustrates the overwhelming power of the new data. Ground-based measurements of the shift in its position were wildly discrepant. The largest reported distance, 102 light-years, was more than three times the smallest, 29 light-years. By contrast, the uncertainty in the actual distance of 59.1 light-years, measured by Hipparcos, has been whittled down to about one per cent, or roughly half a light-year. Please note that as from 1 August 1996 you can access the ESA Space Science Newsletter on Internet, either via the home page : http://www.esrin.esa.it or directly on : http://www.estec.esa.nl/spdwww/h2000/html/snlmain.htm