Content. Chapter 4 Functions Basic concepts on real functions 62. Credits 11


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1 Content Credits 11 Chapter 1 Arithmetic Refresher Algebra 14 Real Numbers 14 Real Polynomials Equations in one variable 21 Linear Equations 21 Quadratic Equations Exercises 28 Chapter 2 Linear systems Definitions Methods for solving linear systems 34 Solving by substitution 34 Solving by elimination Exercises 39 Chapter 3 Trigonometry Angles Triangles Right Triangle Unit Circle 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 Pairs of Angles Sum Identities Inverse Trigonometric Functions Exercises 59 Chapter 4 Functions Basic concepts on real functions 62
2 6 MULTIMEDIA MATHS 4.2 Polynomial functions 63 Linear functions 63 Quadratic functions Intersecting functions Trigonometrical functions 69 Elementary sine function 69 Generalized sine function Inverse trigonometrical functions Exercises 76 Chapter 5 The Golden Section The Golden Number The Golden Section 82 The Golden Triangle 82 The Golden Rectangle 83 The Golden Spiral 84 The Golden Pentagon 86 The Golden Ellipse Golden arithmetics 87 Golden Identities 87 The Fibonacci Numbers The Golden Section worldwide Exercises 93 Chapter 6 Co ordinate systems Cartesian coordinates Parametric curves Polar coordinates Polar curves 102 A polar superformula Exercises 105 Chapter 7 Vectors The concept of a vector 108 Vectors as arrows 108 Vectors as arrays 109 Free Vectors 112 Base Vectors Addition of vectors 112 Vectors as arrows 113
3 CONTENT 7 Vectors as arrays 113 Vector addition summarized 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 Dot product 120 Definition 120 Angle between vectors 121 Orthogonality 123 Vector components in 3D Cross product 126 Definition 126 Parallelism Normal vectors Exercises 131 Chapter 8 Parameters Parametric equations Vector equation of a line Intersecting straight lines Vector equation of a plane Exercises 145 Chapter 9 Collision detection Collision detection and frame rate Collision detection using circles and spheres 149 Circles and spheres 149 Intersecting line and circle 151 Intersecting circles and spheres 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
4 8 MULTIMEDIA MATHS 9.4 Exercises 165 Chapter 10 Matrices The concept of a matrix Determinant of a square matrix Addition of matrices Scalar multiplication of matrices Transpose of a matrix Dot product of matrices 174 Introduction 174 Condition 176 Definition 176 Properties 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 The Fibonacci operator Exercises 189 Chapter 11 Linear transformations Translation Scaling Rotation 200 Rotation in 2D 200 Rotation in 3D Reflection Shearing 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 Conventions Exercises 220
5 CONTENT 9 Chapter 12 Hyp ercomplex numb ers Complex numbers Complex number arithmetics 227 Complex conjugate 227 Addition and subtraction 228 Multiplication 229 Division Complex numbers and transformations Complex continuation of the Fibonacci numbers 235 Integer Fibonacci numbers 235 Complex Fibonacci numbers Quaternions Quaternion arithmetics 238 Addition and subtraction 239 Multiplication 239 Quaternion conjugate 241 Inverse quaternion Quaternions and rotation Exercises 247 Chapter 13 Fractals 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 Selfsimilarity Fractal dimension 258 Euclidean dimension 258 Hausdorff dimension 258 The concept of a logarithm 259 Illustrations The Mandelbrot and Julia Sets 260 Dynamical systems 260 The Mandelbrot Set 262 The Julia Sets Exercises 268
6 10 MULTIMEDIA MATHS Chapter 14 Bezier curves Vector equation of segments 272 Linear Bezier segment 272 Quadratic Bezier segment 273 Cubic Bezier segment 274 Bezier segments of higher degree De Casteljau algorithm Bezier curves 278 Concatenation 278 Linear transformations 280 Illustrations Matrix representation 282 Linear Bezier segment 282 Quadratic Bezier segment 283 Cubic Bezier segment Bsplines 286 Cubic Bsplines 286 Matrix representation 287 De Boor s algorithm 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
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