Maths for Map Makers

Transcription

1 SUB Gottingen A 2003 Maths for Map Makers by Arthur Allan Whittles Publishing

2 Contents /v Chapter 1 Numbers and Calculation '" ter2 The language of mathematics Numbers and your calculator Numbers in exponent form Integers: sequence and series Real or decimal numbers Significant figures Truncated and rounded numbers Operators and stores Reciprocals Powers Factorials Indices in multiplication Indices in division Logarithms The base of natural logarithms e Division using logarithms Powers by logarithms Arithmetic progression Polynomials Sum of the squares of the first n integers Ratios Inequalities The binomial theorem Pascal's triangle General case of binomial theorem Special case of binomial series Plane Geometry Planes and straight lines Straight lines Intersecting, normal and parallel lines

3 vi/ MATHS FOR MAP MAKERS 2.4 Reference grid Parallelogram, rectangle, square and rhombus Pythagoras's theorem for a right angled triangle Proof of Pythagoras's theorem The circle Angles Sexagesimal system Sexagesimal time system Centesimal system Radian system 37 Chapter 3 Trigonometry Introduction Functions of angles The cosine and secant function The cosine formula for a plane triangle Angles greater than a right angle Coordinate axes and bearings Direction cosines ^ Other trigonometrical functions The sine rule in triangle ABC The tangent of the angle 0 or tan The inverse functions sin" 1 and tan" Cosecant and cotangent Summary of the trigonometrical functions The cosine and sine of the compound angle (A + B) Expression for cos (A + B) Expressions for sin (A - JB) and cos (A - B) Expression for tan (A + B) Expressions for sin 2A and cos 2A Expressions for sin 3A and cos 3A Expressions for sums and differences in terms of half angles Half angle formulae Hero's formula for the area of a triangle 61 Chapter 4 Plane Coordinates Plane coordinates in map making Cartesian coordinates (x, y) Plane polar coordinates (r, U) Conversion of polar and cartesian coordinates Coordinate differences Mathematical expressions for a straight line Polar equation of a line in a plane Parametric equations of a line in a plane Angle between two lines 75

4 4.10 The equation of a circle Tangent to a circle Length of a tangent to a circle from a given external point Equation of a tangent to a point on a circle Fitting a circle to three tangents 80 Chapter 5 Problems in Three Dimensions Reference systems A three-dimensional model Signs and coordinate differences Directed steps Length of a line Polar to rectangular coordinates in three dimensions Coordinate differences ' Direction cosines in three dimensions Equations of a line in three dimensions The angle between two rays Parallel and perpendicular rays Projection of one ray on another ^ Distance from a point to a ray The common normal to two rays The length of the normal EF Terminals of the common normal EF The equation of a plane The equation of a sphere The equation of a sphere from four known points Fitting to a circle in three dimensions Creation of a stereoscopic image 109 Chapter 6 Areas and Volumes Areas of plane figures Area of a triangle Area of a trapezium Area of a triangle in terms of plane coordinates Extension to polygon Determinants The tetrahedron Volume of the tetrahedron Volume of the tetrahedron from coordinates 124 Chapter 7 Matrices Matrices An introduction to matrix algebra Solution by elimination Matrix notation 136 CONTENTS /VII

5 vin/ MATHS FOR MAP MAKERS 7.5 Matrix multiplication The inverse of a matrix The inverse of a square matrix Inverse of a (2 x 2) matrix by Cramer's rule Singular matrices More matrix multiplication Other matrix operations: addition and subtraction The transpose of a matrix Further operations on matrices Operations on matrix products: reversal rule Some applications of matrix algebra Rotations in three dimensions Structure of the rotation matrix Rotation of solid body about the Oy and Oz axes Reverse case of rotation Orthogonal matrices Coordinate transformations in three dimensions The reverse transformations 161 Chapter 8 Vectors Vectors Vector algebra Triangle ABC Free vectors and position vectors Orthogonal unit vectors Components and direction cosines Length of a vector Multiplication of a vector by a scalar Multiplication of vectors Scalar multiplication or'dot product' Dot products of orthogonal unit vectors i, j, and k Dot product in terms of components Vector multiplication or'cross product' Cross products of orthogonal unit vectors i, j, and k Cross product in terms of components Parallel and orthogonal vectors The scalar triple product Volume in terms of position vectors Coplanarity of vectors 178 Chapter 9 Calculus Calculus" Functions Limits of functions Tangents to curves General case 186

6 9.6 Proof of the general result Multiple problems Second and higher differentials Maclaurin's theorem Differentiation of trigonometrical functions sin x and cos x Derivative of sin x Derivative of cos* The differentials of e* and In x Taylor's series Differentiation of a product Differentiation of a function of a function Quotient rule for differentiation Partial differentiation Total differential Logarithmic partial differentiation Curvature Integral calculus The notation for an integral Geometrical interpretation of integration Integration limits Areas by integration Simpson's rule for approximate integration Properties of the circle Properties of the sphere f The normal distribution function 217 Chapter 10 Conic Sections Conic sections Sections of a cone Circle Parabola Ellipse Equation of an ellipse Area of an ellipse Sum of focal distances Freedom equations of the ellipse Radius of curvature of the ellipse Length of the normal to an ellipse Length of an arc of the ellipse The hyperbola Range difference property 242 Chapter 11 Spherical Trigonometry Introduction The sphere 245 CONTENTS/ix

7 x/ MATHS FOR MAP MAKERS terl The spherical triangle Model of spherical triangle Spherical trigonometrical formulae Proof of the cosine formula Proof of the sine formula Proof of the cotangent formula Spherical excess Navigation and spherical trigonometry Direction at take-off Map projections of a sphere Oblique coordinate system Stereographic projection of a sphere Azimuthal property of the stereographic projection Conformality of the stereographic projection Scale factor in direction QF Scale factor in direction orthogonal to QF Solution of Equations Introduction Linear equations Simultaneous linear equations Cramer's rule for the solution of linear equations Derivation of Cramer's Rule for a (2 x 2) matrix Note on consistency of linear equations Direct solution by LU decomposition Gaussian elimination Decomposition Forward solution Back solution Inverse of N The solution of a quadratic equation Graphic solution of equations Solution of a cubic equation Newton's method of solution of non-linear equations Formal statement of Newton's method References Appendix Useful data Summary of Formulae Index