Tableofcontent

Content

Credits

11

Chapter 1 · Arithmetic Refresher 1.1

1.2

1.3

13

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

14 14 19 21 21 22 28

C h a p t e r 2 · L i n e ar s y s t e m s 2.1 2.2

2.3

31

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

32 34 34 35 39

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

C h a p t e r 4 · Fu n c t i o n s 4.1

Basic concepts on real functions

41

π 4 π 6 π 3

rad rad rad

42 44 48 49 51 52 52 53 53 54 54 57 59

61 62

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

C h a p t e r 6 路 C o or d i n a t e s y s t e m s 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 or 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

63 63 65 67 69 69 69 73 76

79 80 82 82 83 84 86 86 87 87 88 90 93

95 96 96 99 102 103 105

107 108 108 109 112 112 112 113

CONTENT

7.3

7.4

7.5

7.6 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 Par a m e t e r s 8.1 8.2 8.3 8.4 8.5

7

113 113 114 115 115 116 116 117 118 119 120 120 121 123 124 126 126 128 129 131

133

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

134 135 139 141 145

Chapter 9 路 Collision detection

147

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

148 149 149 151 153 156 156 157 159 161 162

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

165

167 168 169 171 173 174 174 174 176 176 177 179 179 179 180 180 181 184 185 187 189

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

191

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

192 197 200 200 202 204 205 208 210 213 214 215 218 219 220

CONTENT

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 Fr 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

223 224 227 227 228 229 231 233 235 235 236 237 238 239 239 241 242 242 247

249 250 251 251 252 253 253 254 258 258 258 259 259 260 260 262 263 268

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

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

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

A n n e x C · C o m p a n i o n we b s i t e

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