Inhoud

Credits 13

Chapte r 1 · Ar i thmet i c Re f re sher 15

1.1 Algebra 16

Real Numbers 16

Real Polynomials 21

1.2 Equations in one variable 23

Linear Equations 23

Quadratic Equations 24

1.3 Exercises 30

Chapte r 2 · L inea r sy s tems 33

2.1 Definitions 34

2.2 Methods for solving linear systems 36

Solving by substitution 36

Solving by elimination 37

2.3 Exercises 41

Chapte r 3 · Tr i gonomet ry 43

3.1 Angles 44

3.2 Triangles 46

3.3 Right Triangle 50

3.4 Unit Circle 51

3.5 Special Angles 53

Trigonometric ratios for an angle of 45°= π

4rad 54

Trigonometric ratios for an angle of 30°= π

6rad 54

Trigonometric ratios for an angle of 60°= π

3rad 55

Overview 55

3.6 Pairs of Angles 56

3.7 Sum Identities 56

3.8 Inverse Trigonometric Functions 59

3.9 Exercises 61

Chapte r 4 · Func t ions 63

4.1 Basic concepts on real functions 64

6 MULT IMEDIA MATHS

4.2 Polynomial functions 65

Linear functions 65

Quadratic functions 67

4.3 Intersecting functions 69

4.4 Trigonometrical functions 71

Elementary sine function 71

Generalized sine function 71

4.5 Inverse trigonometrical functions 75

4.6 Exercises 78

Chapte r 5 · The Go lden Sec t ion 81

5.1 The Golden Number 82

5.2 The Golden Section 84

The Golden Triangle 84

The Golden Rectangle 85

The Golden Spiral 86

The Golden Pentagon 88

The Golden Ellipse 88

5.3 Golden arithmetics 89

Golden Identities 89

The Fibonacci Numbers 90

5.4 The Golden Section worldwide 92

5.5 Exercises 95

Chapte r 6 · Coord ina te sy s tems 97

6.1 Cartesian coordinates 98

6.2 Parametric curves 98

6.3 Polar coordinates 101

6.4 Polar curves 104

A polar superformula 105

6.5 Exercises 107

Chapte r 7 · Vec tor s 109

7.1 The concept of a vector 110

Vectors as arrows 110

Vectors as arrays 111

Free Vectors 114

Base Vectors 114

7.2 Addition of vectors 114

Vectors as arrows 115

INHOUD 7

Vectors as arrays 115

Vector addition summarized 115

7.3 Scalar multiplication of vectors 116

Vectors as arrows 116

Vectors as arrays 117

Scalar multiplication summarized 118

Properties 118

Vector subtraction 119

Decomposition of a plane vector 120

Base vectors defined 121

7.4 Dot product 122

Definition 122

Angle between vectors 123

Orthogonality 125

Vector components in 3D 126

7.5 Cross product 128

Definition 128

Parallelism 130

7.6 Normal vectors 131

7.7 Exercises 133

Chapte r 8 · Parameter s 135

8.1 Parametric equations 136

8.2 Vector equation of a line 137

8.3 Intersecting straight lines 141

8.4 Vector equation of a plane 143

8.5 Exercises 147

Chapte r 9 · Co l l i s i on de tec t i on 149

9.1 Collision detection and frame rate 150

9.2 Collision detection using circles and spheres 151

Circles and spheres 151

Intersecting line and circle 153

Intersecting circles and spheres 155

9.3 Collision detection using vectors 158

Location of a point with respect to other points 158

Altitude to a straight line 159

Altitude to a plane 161

Frame rate issues 163

Location of a point with respect to a polygon 164

8 MULT IMEDIA MATHS

9.4 Exercises 167

Chapte r 10 · Matr i ce s 169

10.1 The concept of a matrix 170

10.2 Determinant of a square matrix 171

10.3 Addition of matrices 173

10.4 Scalar multiplication of matrices 175

10.5 Transpose of a matrix 176

10.6 Dot product of matrices 176

Introduction 176

Condition 178

Definition 178

Properties 179

10.7 Inverse of a matrix 181

Introduction 181

Definition 181

Conditions 182

Row reduction 182

Matrix inversion 183

Inverse of a product 186

Solving systems of linear equations 187

10.8 The Fibonacci operator 189

10.9 Exercises 191

Chapte r 11 · L inea r t r an s fo rmat ions 193

11.1 Translation 194

11.2 Scaling 199

11.3 Rotation 202

Rotation in 2D 202

Rotation in 3D 204

11.4 Reflection 206

11.5 Shearing 207

11.6 Composing transformations 210

2D rotation around an arbitrary center 212

3D scaling about an arbitrary center 215

2D reflection over an axis through the origin 216

2D reflection over an arbitrary axis 217

3D combined rotation 220

11.7 Conventions 221

11.8 Exercises 222

INHOUD 9

Chapte r 12 · Hypercomp lex number s 225

12.1 Complex numbers 226

12.2 Complex number arithmetics 229

Complex conjugate 229

Addition and subtraction 230

Multiplication 231

Division 233

12.3 Complex numbers and transformations 235

12.4 Complex continuation of the Fibonacci numbers 237

Integer Fibonacci numbers 237

Complex Fibonacci numbers 238

12.5 Quaternions 239

12.6 Quaternion arithmetics 240

Addition and subtraction 241

Multiplication 241

Quaternion conjugate 243

Inverse quaternion 244

12.7 Quaternions and rotation 244

12.8 Exercises 249

Chapte r 13 · Frac t a l s 251

13.1 The concept of a fractal 252

The Sierpinski Gasket 253

The Koch Snowflake 253

The Minkowski Island 254

The Cantor set 255

The Pythagoras Tree 255

13.2 Self-similarity 256

13.3 Fractal dimension 260

Euclidean dimension 260

Hausdorff dimension 260

The concept of a logarithm 261

Illustrations 261

13.4 The Mandelbrot and Julia Sets 262

Dynamical systems 262

The Mandelbrot Set 264

The Julia Sets 265

13.5 Exercises 269

Chapte r 14 · Bez ie r 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

Chapte r A · Rea l number s i n computer s 293

A.1 Scientific notation 293

A.2 The decimal computer 293

A.3 Special values 294

Chapte r B · Nota t ions and Conven t ions 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

Chapte r C · Compan ion webs i t e 299

C.1 Interactivities 299

C.2 Solutions 299

Bibliography 300

Index 303

10 MULT IMEDIA MATHS