Tableofcontents

Inhoud

Credits

13

Chapter 1 · Arithmetic Refresher 1.1

1.2

1.3

15

Algebra Real Numbers Real Polynomials Equations in one variable Linear Equations Quadratic Equations Exercises

16 16 21 23 23 24 30

Chapter 2 · Linear systems 2.1 2.2

2.3

33

Definitions Methods for solving linear systems Solving by substitution Solving by elimination Exercises

34 36 36 37 41

C h a p t e r 3 · Tr i g o n o m e t r y 3.1 3.2 3.3 3.4 3.5

3.6 3.7 3.8 3.9

Angles Triangles Right Triangle Unit Circle Special Angles Trigonometric ratios for an angle of 45°= Trigonometric ratios for an angle of 30°= Trigonometric ratios for an angle of 60°= Overview Pairs of Angles Sum Identities Inverse Trigonometric Functions Exercises

Chapter 4 · Functions 4.1

Basic concepts on real functions

43

π 4 π 6 π 3

rad rad rad

44 46 50 51 53 54 54 55 55 56 56 59 61

63 64

6

M U LT I M E D I A M AT H S

4.2

4.3 4.4

4.5 4.6

Polynomial functions Linear functions Quadratic functions Intersecting functions Trigonometrical functions Elementary sine function Generalized sine function Inverse trigonometrical functions Exercises

Chapter 5 路 The Golden Section 5.1 5.2

5.3

5.4 5.5

The Golden Number The Golden Section The Golden Triangle The Golden Rectangle The Golden Spiral The Golden Pentagon The Golden Ellipse Golden arithmetics Golden Identities The Fibonacci Numbers The Golden Section worldwide Exercises

Chapter 6 路 Coordinate systems 6.1 6.2 6.3 6.4 6.5

Cartesian coordinates Parametric curves Polar coordinates Polar curves A polar superformula Exercises

C h a p t e r 7 路 Ve c t o r s 7.1

7.2

The concept of a vector Vectors as arrows Vectors as arrays Free Vectors Base Vectors Addition of vectors Vectors as arrows

65 65 67 69 71 71 71 75 78

81 82 84 84 85 86 88 88 89 89 90 92 95

97 98 98 101 104 105 107

109 110 110 111 114 114 114 115

INHOUD

7.3

7.4

7.5

7.6 7.7

7

Vectors as arrays Vector addition summarized Scalar multiplication of vectors Vectors as arrows Vectors as arrays Scalar multiplication summarized Properties Vector subtraction Decomposition of a plane vector Base vectors defined Dot product Definition Angle between vectors Orthogonality Vector components in 3D Cross product Definition Parallelism Normal vectors Exercises

C h a p t e r 8 路 Pa r a m e t e r s 8.1 8.2 8.3 8.4 8.5

115 115 116 116 117 118 118 119 120 121 122 122 123 125 126 128 128 130 131 133

135

Parametric equations Vector equation of a line Intersecting straight lines Vector equation of a plane Exercises

136 137 141 143 147

Chapter 9 路 Collision detection

149

9.1 9.2

9.3

Collision detection and frame rate Collision detection using circles and spheres Circles and spheres Intersecting line and circle Intersecting circles and spheres Collision detection using vectors Location of a point with respect to other points Altitude to a straight line Altitude to a plane Frame rate issues Location of a point with respect to a polygon

150 151 151 153 155 158 158 159 161 163 164

8

M U LT I M E D I A M AT H S

9.4

Exercises

C h a p t e r 10 路 M a t r i c e s 10.1 10.2 10.3 10.4 10.5 10.6

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 Introduction Condition Definition Properties 10.7 Inverse of a matrix Introduction Definition Conditions Row reduction Matrix inversion Inverse of a product Solving systems of linear equations 10.8 The Fibonacci operator 10.9 Exercises

167

169 170 171 173 175 176 176 176 178 178 179 181 181 181 182 182 183 186 187 189 191

C h a p t e r 11 路 L i n e a r t r a n s f o r m a t i o n s

193

11.1 Translation 11.2 Scaling 11.3 Rotation Rotation in 2D Rotation in 3D 11.4 Reflection 11.5 Shearing 11.6 Composing transformations 2D rotation around an arbitrary center 3D scaling about an arbitrary center 2D reflection over an axis through the origin 2D reflection over an arbitrary axis 3D combined rotation 11.7 Conventions 11.8 Exercises

194 199 202 202 204 206 207 210 212 215 216 217 220 221 222

INHOUD

C h a p t e r 12 路 H y p e r c o m p l e x n u m b e r s 12.1 Complex numbers 12.2 Complex number arithmetics Complex conjugate Addition and subtraction Multiplication Division 12.3 Complex numbers and transformations 12.4 Complex continuation of the Fibonacci numbers Integer Fibonacci numbers Complex Fibonacci numbers 12.5 Quaternions 12.6 Quaternion arithmetics Addition and subtraction Multiplication Quaternion conjugate Inverse quaternion 12.7 Quaternions and rotation 12.8 Exercises

C h a p t e r 13 路 F r a c t a l s 13.1 The concept of a fractal The Sierpinski Gasket The Koch Snowflake The Minkowski Island The Cantor set The Pythagoras Tree 13.2 Self-similarity 13.3 Fractal dimension Euclidean dimension Hausdorff dimension The concept of a logarithm Illustrations 13.4 The Mandelbrot and Julia Sets Dynamical systems The Mandelbrot Set The Julia Sets 13.5 Exercises

9

225 226 229 229 230 231 233 235 237 237 238 239 240 241 241 243 244 244 249

251 252 253 253 254 255 255 256 260 260 260 261 261 262 262 264 265 269

10

M U LT I M E D I A M AT H S

C h a p t e r 14 · B e z i e r c u r v e s 14.1 Vector equation of segments Linear Bezier segment Quadratic Bezier segment Cubic Bezier segment Bezier segments of higher degree 14.2 De Casteljau algorithm 14.3 Bezier curves Concatenation Linear transformations Illustrations 14.4 Matrix representation Linear Bezier segment Quadratic Bezier segment Cubic Bezier segment 14.5 B-splines Cubic B-splines Matrix representation De Boor’s algorithm 14.6 Exercises

Chapter A · Real numbers in computers A.1 Scientific notation A.2 The decimal computer A.3 Special values

Chapter B · Notations and Conventions B.1 Alphabets Latin alphabet Greek alphabet B.2 Mathematical symbols Sets Mathematical symbols Mathematical keywords Numbers

Chapter C · Companion website

271 272 272 273 274 276 277 278 278 280 280 282 282 283 284 286 286 287 289 291

293 293 293 294

295 295 295 295 296 296 297 297 298

299

C.1 Interactivities C.2 Solutions

299 299

Bibliography

300

Index

303