DOCSLIB.ORG

A Treatise on Spherical Trigonometry, and Its Application

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Treatise on Spherical Trigonometry, and Its Application A T REATISE SPHEEICAL T RIGONOMETKY. WORKSY B JOHN G ASEY, ESQ., LLD., F. R.8., FELLOWF O THE ROYAL UNIVERSITY OF IRELAND. Second E dition, Price 3s. A T REATISE ON ELEMENTARY TRIGONOMETRY, With n umerous Examples AND E&uesttfms f or Exammatttm. OKEY T THE EXERCISES IN THE TREATISE ON ELEMENTARY TRIGONOMETRY. Fifth E dition, Revised and Enlarged, Price 3s. 6d., Cloth. A S EQ,UEL TO THE FIRST SIX BOOKS OF THE ELEMENTS OF EUCLID, Containing- a n Easy Introduction to Modern Geometry . With a umerstts Examples. Seventh E dition, Price 4s. 6d. ; or in two parts, each 2s. 6d. THE E LEMENTS OE EUCLID, BOOKS I.-YL, AND PROPOSITIONS I.-XXI. OP BOOK XI. ; Together ' with an Appendix on the Cylinder, Sphere, Cone, &>c. @opdatt$ J tettotatiotts & aumeraus Exercises. Second E dition, Price 6s. AEY K TO THE EXERCISES IN THE FIRST SIX BOOKS OF CASEY'S "ELEMENTS OF EUCLID." Price 7 s. 6d. A T REATISE ON THE ANALYTICAL GEOMETRY OF THE POINT, LINE, CIRCLE, & CONIC SECTIONS, Containing a n Account of its most recent Extensions, Wxih t tttmeratts Examples. Price 7 s. 6d. A T REATISE ON PLANE TRIGONOMETRY, including THE THEORY OF HYPERBOLIC FUNCTIONS. LONDON : L ONGMANS? CO. DUBLIN: HODGES, FTGGIS & CO. A T REATISE ON- SPHERICAL T RIGONOMETRY, ANDTS I APPLICATION- TO GEODESYND A ASTRONOMY, ■WITH BY JOHN C ASEY, LL.D., F.E.S., Fellowf o the Royal University of Ireland; Memberf o the Council of the Royal Irish Academy ; Memberf o the Mathematical Societies of London and France ; Corresponding M ember of the Royal Society of Sciences of Liege; and Professorf o the Higher Mathematics and Mathematical Physics in t he Catholic University of Ireland. DUBLIN: H ODGES, EIGGIS, & CO., GKAFTON-ST. LONDON : L ONGMANS, GREEN, & CO. 1889. DUBLIN I PRINTEDT A THE UNIVERSITY PRESS, BY P ONSONBY AND WELDRICK. PREFACE. The p resent Manual is intended as a Sequel to the Author's Treatise on Plane Trigonometry, and is written on the same plan. An examination of the Table of Contents, or of the Index, will show the scope of the work. It will he seen that, though moderate in size, it contains a large amount of matter, much of which is original. The s ources from which I have obtained information are indicated in the text. The principal are Crelle's Journal " fur die reine und angewandte Mathematik," Berlin, and Nouvelles Annates de Mathimatiques, Paris. The e xamples, which are very numerous (over five hundred) and carefully selected, illustrate every part of the subject. Among them will be found some of the most elegant Theorems in Spherical Geometry and Trigonometry. vi P reface. In t he preparation and arrangement of every part of the work I have received invaluable assistance from Professor Neuberg, of the University of Liege. For this, as well as for similar assistance previously given in the editing of my Plane Trigonometry, I beg to return that gentleman my most grateful acknowledg ments and best thanks. JOHN C ASEY. 86, S outh Circular Road, Dublin. March 25, 1889. CONTENTS. CHAPTER I . SPHERICAL G EOMETRY. PAGE Section I . — Preliminary Propositions and Definitions, Definitions o f Sphere, ,, C entre of Sphere, ...... „ R adius ,, ,, D iameter ,, Curvef o intersection of sphere and plane, 2 fPoles o circles defined, 3 Small c ircles and great circles defined, 3 Intersection o f two spheres, 3 Secondary c ircles defined, . 4 Spherical r adius, 4 Only o ne great circle through two points, not diametrically opposite, 4 Locusf o points of sphere equidistant from two fixed points on sphere, 5 Two g reat circles bisect each other, 5 Anglef o intersection of two circles on sphere defined, . 6 ,, o f two great circles is equal to the inclina tionf o their planes, 6 To f ind the radius of a solid sphere, . • 6 Analogy b etween the geometry of the sphere and the plane, . 7 Exercises I ., 8 Section I I. — Spherical Triangles, 9 Spherical t riangle defined, 9 Correspondence b etween a solid angle and a spherical triangle, . 9 Lune d efined, 10 Colunar t riangles, 10 viii C ontents. PAGE. Antipodal t riangles, , 10 Any t wo sides of a spherical triangle are greater than the third, and t he sum of the three sides is less than a great circle, . 10 Positive a nd negative poles distinguished, 11 Polar t riangles defined, .11 Eelation b etween polar triangles, 12 Sumf o angles of a spherical triangle, 12 Sumf o external angles of a spherical polygon, . .13 Spherical e xcess defined, 13 Antipodal t riangles are equal in area, 13 nCases i which two spherical triangles on the same sphere have all t heir corresponding elements respectively equal, . 14 Properties o f isosceles spherical triangles, 15 Areaf o a lune, .15 Girard's t heorem. — The area of a spherical triangle, ... 16 Eatiof o area of a spherical triangle to area of a great circle, . 17 Areaf o spherical polygon, 17 Exercises I I., .17 Diametrical t riangles defined, 18 Spherical p arallelograms defined, 18 CHAPTEE I I. FORMULAE C ONNECTING THE SIDES AND ANGLES OF A SPHERICAL T RIANGLE. Three c lasses of formulae, 19 Section I . — First class, . 19 .Case I — Three sides and an angle, 19 Exercises I II., 22 First S taudtian of a spherical triangle, . ... .22 Second S taudtian of a spherical triangle, 22 Normsf o the sides and angles of a spherical triangle, . .23 I.Case I — Two sides and their opposite angles, .... 24 The s ines of the sides of a spherical triangle are proportional to the s ines of their opposite angles, 24 Exercises I V., 26 Case I II. — Two sides and two angles, one of which is contained by t he sides, 27 Exercises V ., 28 CaseV. I — Three angles and a side, . t . .29 Exercises V I., 31 Substitutions m ade in order to pass from a spherical triangle to its p olar triangle, 32 Contents. i x PAGE. Section I I. — First class continued, 32 The r ight-angled triangle, 32 Comparison o f formulas for right-angled triangles, plane an spherical, 3 4 Napier's r ules, 35 Exercises V II., 36 Quadrantal t riangles, 38 Section I II. — Second Class, 39 Formulae c ontaining five elements, 39 Napier's a nalogies, 39 Exercises V III., 40 V.Section I — Third class, 40 Delamhre's a nalogies, 40 Historical s ketch of Delamhre's analogies, 41 Reidt's a nalogies, 41 Other a pplications of Delamhre's analogies, . .43 Lhuilier's t heorem, 44 The L huilierian function X, 44 The d ouble value of i, .^ ;. 44 Exercises I X., 45 Breitschneider's a nalogies, 47 CHAPTER I II. SOLUTIONF O SPHERICAL TRIANGLES. Preliminary o hservations, 48 Usef o auxiliary angles, 48 Section I . — The right-angled triangle, 49 Six c ases of right-angled triangles, 49 First c ase. — Being given c and a, to calculate b, A, B, . 49 Typef o the calculation, 50 Exercises X ., 50 Second c ase. — Being given c, A, to calculate b, a, By . 50 Typef o the calculation, 51 Exercises X I., 51 Third c ase. — Being given «, b, to calculate A, B, c, . 51 Exercises X II., . 52 Fourth c ase. — Being given a, A, to find c, b, B, . 52 Exercises X III., 53 Fifth c ase. — Being given c, B, to find b, A, c, . 53 Exercises X IV., 53 Sixth c ase. — Being given A} B, to find a, b, e, . .53 Exercises X V., 54 fSolution o quadrantal triangles, . • . .54 x C ontents. PAGE. Section I I. — Oblique -angled triangles, . .54 Three p airs of cases of oblique-angled triangles, . ... 54 First p air of cases, 55 Being g iven the three sides a> b> c, to calculate the angles, . 55 Typef o the calculation, 55 Exercises X VI., 56 Methodf o calculation when only one angle is required, . 56 Being g iven the three angles A, B, G to calculate the sides, . 56 Exercises X VII., 57 Methodf o calculation when only one side is required, . 57 Second p air of cases, 58 Being g iven two sides a, b, and the angle opposite to one of them, to c alculate the remaining parts, 58 The a mbiguous case, 58 Reidt's a nalogies, advantages of, in calculation, . 59 This c ase solved by dividing the triangle into two right-angled triangles, 6 0 Exercises X VIII., 60 Oiven t wo angles, Ay B, and the side opposite to one of them to solve t he triangle, 61 Third p air of cases, 61 Being g iven the sides a, b and the contained angle C to calculate A,, B e, 61 Typef o the calculation, . 61 Exercises X IX., 62 Being g iven two angles A, B and the adjacent side c to find a, b, C, 63 Cauchy's m ethod of solving the various cases of oblique-angled triangles, 6 3 Exercises X X., 65 Briinnow's s eries, ......*.. 65 CHAPTEE I V. VARIOUS A PPLICATIONS. Section I . — Theory of transversals, 67 Ratiof o section of an arc, 67 Anharmonic r atio of four points, on an arc of a great circle, . 68 Ratiof o section of an angle, . .68 Anharmonic r atio of four great circles passing through the same point, 6 9 Properties o f a great circle intersecting the sides of a triangle, . 69 Medians o f a spherical triangle, 70 The a ltitudes, 70 The c ommon orthocentre of two supplemental triangles, . 70 Contents. x i PAGE.
Load more

Recommended publications
  • 1 Chapter 3 Plane and Spherical Trigonometry 3.1
    1 CHAPTER 3 PLANE AND SPHERICAL TRIGONOMETRY 3.1 Introduction It is assumed in this chapter that readers are familiar with the usual elementary formulas encountered in introductory trigonometry. We start the chapter with a brief review of the solution of a plane triangle. While most of this will be familiar to readers, it is suggested that it be not skipped over entirely, because the examples in it contain some cautionary notes concerning hidden pitfalls. This is followed by a quick review of spherical coordinates and direction cosines in three- dimensional geometry. The formulas for the velocity and acceleration components in two- dimensional polar coordinates and three-dimensional spherical coordinates are developed in section 3.4. Section 3.5 deals with the trigonometric formulas for solving spherical triangles. This is a fairly long section, and it will be essential reading for those who are contemplating making a start on celestial mechanics. Sections 3.6 and 3.7 deal with the rotation of axes in two and three dimensions, including Eulerian angles and the rotation matrix of direction cosines. Finally, in section 3.8, a number of commonly encountered trigonometric formulas are gathered for reference. 3.2 Plane Triangles . This section is to serve as a brief reminder of how to solve a plane triangle. While there may be a temptation to pass rapidly over this section, it does contain a warning that will become even more pertinent in the section on spherical triangles. Conventionally, a plane triangle is described by its three angles A, B, C and three sides a, b, c, with a being opposite to A, b opposite to B, and c opposite to C.
    [Show full text]
  • Geodesy. Application of the Theory of Least Squares to the Adjustment Of
    SI No Serial ^) DEPARTME]S'T OF COMMERCE 0. S. COAST AND GEODETIC SURVEY it E. LESTER JONES, Supei^intendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION RY OSCAR S. ADAM^ COMPUTER UNITED STATES COAST AND GEODETIC SURJ^Y No. 28 Special Publication { WASHINGTON GOVERNMENT I'RINTING OFFICE 1915 Serial No. 9 DEPARTMEISTT OF COMMERCE U. S. COAST AND GEODETIC SURVEY E. LESTER JONES, Superintendent GEODESY APPLICATION OF THE THEORY OF LEAST SQUARES TO THE ADJUSTMENT OF TRIANGULATION BY OSCAR S. ADAMS COMPUTER UNITED STATES COAST AND GEODETIC SURVEY Special Publication No. 28 WASHINGTON GOVERNMENT PRINTING OFFICE 1815 ADDITIONAL COPIES OF THIS PUBLICATION MAY BE PROCURED FROM THE SUPERINTENDENT OF DOCUMENTS GOVERNMENT PRINTING OFFICE WASHINGTON, D. C. AT 25 CENTS PER COPY QB 32/ CONTENTS Page. General statement 7 Station adjustment 7 Observed angles 8 List of directions 8 Condition equations , 9 Formation of normal equations by differentiation 9 Correlate equations 11 Formation of normal equations 11 Normal equations 12 Discussion of method of solution of normal equations 12 Solution of normal equations 13 Back solution 13 Computation of corrections 13 Adjustment of a quadrilateral 14 General statement 14 Lists of directions 16 Figure 16 Angle equations 17 Side equation 17 Formation of normal equations by differentiation 17 Correlate equations 18 Normal equations 18 Solution of normal equations 19 Back solution 19 Computation of corrections 19 Adjustment of a quadrilateral by the use of two angle and two
    [Show full text]
  • Hyperbolic Trigonometry in Two-Dimensional Space-Time Geometry
    Hyperbolic trigonometry in two-dimensional space-time geometry F. Catoni, R. Cannata, V. Catoni, P. Zampetti ENEA; Centro Ricerche Casaccia; Via Anguillarese, 301; 00060 S.Maria di Galeria; Roma; Italy January 22, 2003 Summary.- By analogy with complex numbers, a system of hyperbolic numbers can be intro- duced in the same way: z = x + hy; h2 = 1 x, y R . As complex numbers are linked to the { ∈ } Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representa- tion, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalise the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry. PACS 03 30 - Special Relativity PACS 02.20. Hj - Classical groups and Geometries 1 Introduction Complex numbers are strictly related to the Euclidean geometry: indeed their invariant (the module) arXiv:math-ph/0508011v1 3 Aug 2005 is the same as the Pythagoric distance (Euclidean invariant) and their unimodular multiplicative group is the Euclidean rotation group. It is well known that these properties allow to use complex numbers for representing plane vectors. In the same way hyperbolic numbers, an extension of complex numbers [1, 2] defined as z = x + hy; h2 =1 x, y R , { ∈ } are strictly related to space-time geometry [2, 3, 4]. Indeed their square module given by1 z 2 = 2 2 | | zz˜ x y is the Lorentz invariant of two dimensional special relativity, and their unimodular multiplicative≡ − group is the special relativity Lorentz group [2].
    [Show full text]
  • Spherical Trigonometry
    Spherical trigonometry 1 The spherical Pythagorean theorem Proposition 1.1 On a sphere of radius R, any right triangle 4ABC with \C being the right angle satisfies cos(c=R) = cos(a=R) cos(b=R): (1) Proof: Let O be the center of the sphere, we may assume its coordinates are (0; 0; 0). We may rotate −! the sphere so that A has coordinates OA = (R; 0; 0) and C lies in the xy-plane, see Figure 1. Rotating z B y O(0; 0; 0) C A(R; 0; 0) x Figure 1: The Pythagorean theorem for a spherical right triangle around the z axis by β := AOC takes A into C. The edge OA moves in the xy-plane, by β, thus −−!\ the coordinates of C are OC = (R cos(β);R sin(β); 0). Since we have a right angle at C, the plane of 4OBC is perpendicular to the plane of 4OAC and it contains the z axis. An orthonormal basis −−! −! of the plane of 4OBC is given by 1=R · OC = (cos(β); sin(β); 0) and the vector OZ := (0; 0; 1). A rotation around O in this plane by α := \BOC takes C into B: −−! −−! −! OB = cos(α) · OC + sin(α) · R · OZ = (R cos(β) cos(α);R sin(β) cos(α);R sin(α)): Introducing γ := \AOB, we have −! −−! OA · OB R2 cos(α) cos(β) cos(γ) = = : R2 R2 The statement now follows from α = a=R, β = b=R and γ = c=R. ♦ To prove the rest of the formulas of spherical trigonometry, we need to show the following.
    [Show full text]
  • 7 TRIGONOMETRY Objectives in This Chapter You Will Learn About • Revision of Solution of Triangles
    7 TRIGONOMETRY Objectives In this chapter you will learn about • Revision of solution of triangles. • Solving problems in 2–dimensions and in 3-dimensions by constructing and interpreting models. 1 | P a g e 7.1 Revision In previous grades, we dealt with the types of triangles named according to side and angles. In trigonometry, we are able to find the unknown sides and angles when given the magnitudes of the minimum required number of angles and sides. We looked at two main types of triangles: • Right angled triangles. • Non-right angled triangles Right angled triangles When a right angled triangle is given, the following may be useful: • Pythagoras theorem • Definitions of Trigonometric ratios Non-right angled triangles • Sine rule • Cosine rule • Area rule NOTE: • Not all triangles that you need to solve are right-angled triangles. In grade 11 you learned about rules which can be used to solve non-right-angled triangles. • Theorems/axioms involving lines, triangles, quadrilateral and circles are also useful. 7.1.1 The Sine Rule The sine rule expresses the relationship between the two of the sides of the triangles and two of the angles. In any triangle, a is the side opposite angle A, b is the side opposite angle B and c is the side opposite angle C, then the lengthof any side the length of any other side = the sine of the angle opposite that side the sine of the angle opposite that side That is: 2 | P a g e In ABC, a b c = = sin A sinB sinC This can also be written as: sin A sinB sinC = = a b c The sine rule can be used when the following information about a triangle is given: • Two sides and an angle opposite to one of the two sides • One side and any two angles Worked Example 1 Solve the triangle given below.
    [Show full text]
  • The Sphere of the Earth Description and Technical Manual
    The Sphere of the Earth Description and Technical Manual December 20, 2012 Daniel Ramos ∗ MMACA (Museu de Matem`atiquesde Catalunya) 1 The exhibit The exhibit we are presenting explores the science of cartography and the geometry of the sphere. Althoug this subject has been widely treated on science exhibitions and fairs, our proposal aims to be a more comprehensive and complete treatment on the subject, as well as to be a fully open source, documented material that both museographers and public can experience, learn and modify. The key concept comes from the problem of representing the spherical surface of the Earth onto a flat map. Studying this problem is the subject of cartography, and has been an important mathematical problem in History (navigation, position, frontiers, land ownership...). An essential theorem in Geometry (Gauss' Egregium theorem) ensures that there is no perfect map, that is, there is no way of repre- senting the Earth keeping distances at scale. However, this is exactly what makes cartography a discipline: developing several different maps that try to solve well enough the problem of representing the Earth. A common activity is comparing an Earth globe with a (Mercator) map, observ- ing that distances are not preserved. Our module explores far beyond that. We present: • Several different maps, currently 6 map projections are used, each one featuring different special properties. Printed at nominal scale 1:1 of a model globe. • The script programs that generate these pictures. Generating a map takes less than 50 lines of code, anyone could generate his own. ∗E-mail: [email protected] 1 • A collection of tools and models: a flexible ruler to be used over the globe, plane and spherical protractors, models showing longitude and latitude..
    [Show full text]
  • Spherical Trigonometry Through the Ages
    School of Mathematics and Statistics Newcastle University MMath Mathematics Report Spherical Trigonometry Through the Ages Author: Supervisor: Lauren Roberts Dr. Andrew Fletcher May 1, 2015 Contents 1 Introduction 1 2 Geometry on the Sphere Part I 4 2.1 Cross-sections and Measurements . .4 2.2 Triangles on the Sphere . .7 2.2.1 What is the Largest Possible Spherical Triangle? . .7 2.2.2 The Sum of Spherical Angles . .9 3 Menelaus of Alexandria 12 3.1 Menelaus' Theorem . 13 3.2 Astronomy and the Menelaus Configuration: The Celestial Sphere . 16 4 Medieval Islamic Mathematics 18 4.1 The Rule of Four Quantities . 18 4.2 Finding the Qibla . 20 4.3 The Spherical Law of Sines . 22 5 Right-Angled Triangles 25 5.1 Pythagoras' Theorem on the Sphere . 25 5.2 The Spherical Law of Cosines . 27 6 Geometry on the Sphere Part II 30 6.1 Areas of Spherical Polygons . 30 6.2 Euler's Polyhedral Formula . 33 7 Quaternions 35 7.1 Some Useful Properties . 35 7.1.1 The Quaternion Product . 36 7.1.2 The Complex Conjugate and Inverse Quaternions . 36 1 CONTENTS 2 7.2 Rotation Sequences . 37 7.2.1 3-Dimensional Representation . 39 7.3 Deriving the Spherical Law of Sines . 39 8 Conclusions 42 Bibliography 44 Abstract The art of applying trigonometry to triangles constructed on the surface of the sphere is the branch of mathematics known as spherical trigonometry. It allows new theorems relating the sides and angles of spherical triangles to be derived. The history and development of spherical trigonometry throughout the ages is discussed.
    [Show full text]
  • Arxiv:1409.4736V1 [Math.HO] 16 Sep 2014
    ON THE WORKS OF EULER AND HIS FOLLOWERS ON SPHERICAL GEOMETRY ATHANASE PAPADOPOULOS Abstract. We review and comment on some works of Euler and his followers on spherical geometry. We start by presenting some memoirs of Euler on spherical trigonometry. We comment on Euler's use of the methods of the calculus of variations in spherical trigonometry. We then survey a series of geometrical resuls, where the stress is on the analogy between the results in spherical geometry and the corresponding results in Euclidean geometry. We elaborate on two such results. The first one, known as Lexell's Theorem (Lexell was a student of Euler), concerns the locus of the vertices of a spherical triangle with a fixed area and a given base. This is the spherical counterpart of a result in Euclid's Elements, but it is much more difficult to prove than its Euclidean analogue. The second result, due to Euler, is the spherical analogue of a generalization of a theorem of Pappus (Proposition 117 of Book VII of the Collection) on the construction of a triangle inscribed in a circle whose sides are contained in three lines that pass through three given points. Both results have many ramifications, involving several mathematicians, and we mention some of these developments. We also comment on three papers of Euler on projections of the sphere on the Euclidean plane that are related with the art of drawing geographical maps. AMS classification: 01-99 ; 53-02 ; 53-03 ; 53A05 ; 53A35. Keywords: spherical trigonometry, spherical geometry, Euler, Lexell the- orem, cartography, calculus of variations.
    [Show full text]
  • 4 Trigonometry and Complex Numbers
    4 Trigonometry and Complex Numbers Trigonometry developed from the study of triangles, particularly right triangles, and the relations between the lengths of their sides and the sizes of their angles. The trigono- metric functions that measure the relationships between the sides of similar triangles have far-reaching applications that extend far beyond their use in the study of triangles. Complex numbers were developed, in part, because they complete, in a useful and ele- gant fashion, the study of the solutions of polynomial equations. Complex numbers are useful not only in mathematics, but in the other sciences as well. Trigonometry Most of the trigonometric computations in this chapter use six basic trigonometric func- tions The two fundamental trigonometric functions, sine and cosine, can be de¿ned in terms of the unit circle—the set of points in the Euclidean plane of distance one from the origin. A point on this circle has coordinates +frv w> vlq w,,wherew is a measure (in radians) of the angle at the origin between the positive {-axis and the ray from the ori- gin through the point measured in the counterclockwise direction. The other four basic trigonometric functions can be de¿ned in terms of these two—namely, vlq { 4 wdq { @ vhf { @ frv { frv { frv { 4 frw { @ fvf { @ vlq { vlq { 3 ?w? For 5 , these functions can be found as a ratio of certain sides of a right triangle that has one angle of radian measure w. Trigonometric Functions The symbols used for the six basic trigonometric functions—vlq, frv, wdq, frw, vhf, fvf—are abbreviations for the words cosine, sine, tangent, cotangent, secant, and cose- cant, respectively.You can enter these trigonometric functions and many other functions either from the keyboard in mathematics mode or from the dialog box that drops down when you click or choose Insert + Math Name.
    [Show full text]
  • GEOMETRIC ANALYSIS of an OBSERVER on a SPHERICAL EARTH and an AIRCRAFT OR SATELLITE Michael Geyer
    GEOMETRIC ANALYSIS OF AN OBSERVER ON A SPHERICAL EARTH AND AN AIRCRAFT OR SATELLITE Michael Geyer S π/2 - α - θ d h α N U Re Re θ O Federico Rostagno Michael Geyer Peter Mercator nosco.com Project Memorandum — September 2013 DOT-VNTSC-FAA-13-08 Prepared for: Federal Aviation Administration Wake Turbulence Research Program DOT/RITA Volpe Center TABLE OF CONTENTS 1. INTRODUCTION...................................................................................................... 1 1.1 Basic Problem and Solution Approach........................................................................................1 1.2 Vertical Plane Formulation ..........................................................................................................2 1.3 Spherical Surface Formulation ....................................................................................................3 1.4 Limitations and Applicability of Analysis...................................................................................4 1.5 Recommended Approach to Finding a Solution.........................................................................5 1.6 Outline of this Document ..............................................................................................................6 2. MATHEMATICS AND PHYSICS BASICS ............................................................... 8 2.1 Exact and Approximate Solutions to Common Equations ........................................................8 2.1.1 The Law of Sines for Plane Triangles.........................................................................................8
    [Show full text]
  • An Axiom of True Courses Calculation in Great Circle Navigation
    Journal of Marine Science and Engineering Communication An Axiom of True Courses Calculation in Great Circle Navigation Mate Baric 1, David Brˇci´c 2,* , Mate Kosor 1 and Roko Jelic 1 1 Maritime Department, University of Zadar, 23000 Zadar, Croatia; [email protected] (M.B.); [email protected] (M.K.); [email protected] (R.J.) 2 Faculty of Maritime Studies, University of Rijeka, 51000 Rijeka, Croatia * Correspondence: [email protected] Abstract: Based on traditional expressions and spherical trigonometry, at present, great circle naviga- tion is undertaken using various navigational software packages. Recent research has mainly focused on vector algebra. These problems are calculated numerically and are thus suited to computer-aided great circle navigation. However, essential knowledge requires the navigator to be able to calculate navigation parameters without the use of aids. This requirement is met using spherical trigonometry functions and the Napier wheel. In addition, to facilitate calculation, certain axioms have been developed to determine a vessel’s true course. These axioms can lead to misleading results due to the limitations of the trigonometric functions, mathematical errors, and the type of great circle navigation. The aim of this paper is to determine a reliable trigonometric function for calculating a vessel’s course in regular and composite great circle navigation, which can be used with the proposed axioms. This was achieved using analysis of the trigonometric functions, and assessment of their impact on the vessel’s calculated course and established axioms. Citation: Baric, M.; Brˇci´c,D.; Kosor, Keywords: great circle; navigation; axiom M.; Jelic, R. An Axiom of True Courses Calculation in Great Circle Navigation.
    [Show full text]
  • Spherical Trigonometry and Navigational Calculations
    1 Spherical Trigonometry and Navigational Calculations Badar Abbas, Student, EME College, [email protected] In the 13th century, Nasir al-Din al Tusi (1201–74) and al- Abstract—This paper discusses the fundamental concepts in Battani, continued to develop spherical trigonometry. Tusi was spherical trigonometry and its application to the navigational the first (c. 1250) to write a work on trigonometry calculation. It briefly discusses the history of the spherical independently of astronomy. The final major development in trigonometry. It describes the commonly used terms in navigation and spherical trigonometry. It specifically concentrates on great- classical trigonometry was the invention of logarithms by the circle and dead-reckoning navigation. It also outlines the various Scottish mathematician John Napier in 1614 that greatly functions available in the Mapping Toolbox of MATLAB for such facilitated the art of numerical computation—including the navigational calculations. compilation of trigonometry tables [4]. Index Terms— Dead-reckoning, Geodesic, Great-circle, Navigation, Spherical Trigonometry. III. NAVIGATIONAL TERMINOLOGY Although the Earth is very round, in fact, it is a flattened I. INTRODUCTION sphere or spheroid with values for the radius of curvature of 6336 km at the equator and 6399 km at the poles. AVIGATION is the process of planning, recording, Approximating the earth as a sphere with a radius of 6370 km and controlling the movement of a craft or vehicle from N results in an actual error of up to about 0.5% [5]. The one location to another. The word derives from the Latin roots flattening of the ellipsoid is ~1/300 (1/298.257222101 is the navis (“ship”) and agere (“to move or direct”).
    [Show full text]