Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter 2 Linear systems 31 2.1 Definitions 32 2.2 Methods for solving linear systems 34 Solving by substitution 34 Solving by elimination 35 2.3 Exercises 39 Chapter 3 Trigonometry 41 3.1 Angles 42 3.2 Triangles 44 3.3 Right Triangle 48 3.4 Unit Circle 49 3.5 Special Angles 51 Trigonometric ratios for an angle of 45 = π 4 rad 52 Trigonometric ratios for an angle of 30 = π 6 rad 52 Trigonometric ratios for an angle of 60 = π 3 rad 53 Overview 53 3.6 Pairs of Angles 54 3.7 Sum Identities 54 3.8 Inverse Trigonometric Functions 57 3.9 Exercises 59 Chapter 4 Functions 61 4.1 Basic concepts on real functions 62
6 MULTIMEDIA MATHS 4.2 Polynomial functions 63 Linear functions 63 Quadratic functions 65 4.3 Intersecting functions 67 4.4 Trigonometrical functions 69 Elementary sine function 69 Generalized sine function 69 4.5 Inverse trigonometrical functions 73 4.6 Exercises 76 Chapter 5 The Golden Section 79 5.1 The Golden Number 80 5.2 The Golden Section 82 The Golden Triangle 82 The Golden Rectangle 83 The Golden Spiral 84 The Golden Pentagon 86 The Golden Ellipse 86 5.3 Golden arithmetics 87 Golden Identities 87 The Fibonacci Numbers 88 5.4 The Golden Section worldwide 90 5.5 Exercises 93 Chapter 6 Co ordinate systems 95 6.1 Cartesian coordinates 96 6.2 Parametric curves 96 6.3 Polar coordinates 99 6.4 Polar curves 102 A polar superformula 103 6.5 Exercises 105 Chapter 7 Vectors 107 7.1 The concept of a vector 108 Vectors as arrows 108 Vectors as arrays 109 Free Vectors 112 Base Vectors 112 7.2 Addition of vectors 112 Vectors as arrows 113
CONTENT 7 Vectors as arrays 113 Vector addition summarized 113 7.3 Scalar multiplication of vectors 114 Vectors as arrows 115 Vectors as arrays 115 Scalar multiplication summarized 116 Properties 116 Vector subtraction 117 Decomposition of a plane vector 118 Base vectors defined 119 7.4 Dot product 120 Definition 120 Angle between vectors 121 Orthogonality 123 Vector components in 3D 124 7.5 Cross product 126 Definition 126 Parallelism 128 7.6 Normal vectors 129 7.7 Exercises 131 Chapter 8 Parameters 133 8.1 Parametric equations 134 8.2 Vector equation of a line 135 8.3 Intersecting straight lines 139 8.4 Vector equation of a plane 141 8.5 Exercises 145 Chapter 9 Collision detection 147 9.1 Collision detection and frame rate 148 9.2 Collision detection using circles and spheres 149 Circles and spheres 149 Intersecting line and circle 151 Intersecting circles and spheres 153 9.3 Collision detection using vectors 156 Location of a point with respect to other points 156 Altitude to a straight line 157 Altitude to a plane 159 Frame rate issues 161 Location of a point with respect to a polygon 162
8 MULTIMEDIA MATHS 9.4 Exercises 165 Chapter 10 Matrices 167 10.1 The concept of a matrix 168 10.2 Determinant of a square matrix 169 10.3 Addition of matrices 171 10.4 Scalar multiplication of matrices 173 10.5 Transpose of a matrix 174 10.6 Dot product of matrices 174 Introduction 174 Condition 176 Definition 176 Properties 177 10.7 Inverse of a matrix 179 Introduction 179 Definition 179 Conditions 180 Row reduction 180 Matrix inversion 181 Inverse of a product 184 Solving systems of linear equations 185 10.8 The Fibonacci operator 187 10.9 Exercises 189 Chapter 11 Linear transformations 191 11.1 Translation 192 11.2 Scaling 197 11.3 Rotation 200 Rotation in 2D 200 Rotation in 3D 202 11.4 Reflection 204 11.5 Shearing 205 11.6 Composing transformations 208 2D rotation around an arbitrary center 210 3D scaling about an arbitrary center 213 2D reflection over an axis through the origin 214 2D reflection over an arbitrary axis 215 3D combined rotation 218 11.7 Conventions 219 11.8 Exercises 220
CONTENT 9 Chapter 12 Hyp ercomplex numb ers 223 12.1 Complex numbers 224 12.2 Complex number arithmetics 227 Complex conjugate 227 Addition and subtraction 228 Multiplication 229 Division 231 12.3 Complex numbers and transformations 233 12.4 Complex continuation of the Fibonacci numbers 235 Integer Fibonacci numbers 235 Complex Fibonacci numbers 236 12.5 Quaternions 237 12.6 Quaternion arithmetics 238 Addition and subtraction 239 Multiplication 239 Quaternion conjugate 241 Inverse quaternion 242 12.7 Quaternions and rotation 242 12.8 Exercises 247 Chapter 13 Fractals 249 13.1 The concept of a fractal 250 The Sierpinski Gasket 251 The Koch Snowflake 251 The Minkowski Island 252 The Cantor set 253 The Pythagoras Tree 253 13.2 Self-similarity 254 13.3 Fractal dimension 258 Euclidean dimension 258 Hausdorff dimension 258 The concept of a logarithm 259 Illustrations 259 13.4 The Mandelbrot and Julia Sets 260 Dynamical systems 260 The Mandelbrot Set 262 The Julia Sets 263 13.5 Exercises 268
10 MULTIMEDIA MATHS Chapter 14 Bezier curves 271 14.1 Vector equation of segments 272 Linear Bezier segment 272 Quadratic Bezier segment 273 Cubic Bezier segment 274 Bezier segments of higher degree 276 14.2 De Casteljau algorithm 277 14.3 Bezier curves 278 Concatenation 278 Linear transformations 280 Illustrations 280 14.4 Matrix representation 282 Linear Bezier segment 282 Quadratic Bezier segment 283 Cubic Bezier segment 284 14.5 B-splines 286 Cubic B-splines 286 Matrix representation 287 De Boor s algorithm 289 14.6 Exercises 291 Annex A Real numb ers in computers 293 A.1 Scientific notation 293 A.2 The decimal computer 293 A.3 Special values 294 Annex B Notations and Conventions 295 B.1 Alphabets 295 Latin alphabet 295 Greek alphabet 295 B.2 Mathematical symbols 296 Sets 296 Mathematical symbols 297 Mathematical keywords 297 Numbers 298 Annex C Companion website 299 C.1 Interactivities 299 C.2 Solutions 299 Bibliography 300 Index 303