# Essential Mathematics for Computer Graphics fast

Springer Science & Business Media, 2001 - Computers - 229 pages
Baffled by maths? Then don't give up hope.John Vince will show you how to understand many of the mathematical ideas used in computer animation, virtual reality, CAD, and other areas of computer graphics.In ten chapters you will rediscover - and hopefully discover for the first time a new way of understanding - the mathematical techniques required to solve problems and design computer programs for computer graphic applications. Each chapter explores a specific mathematical topic and takes you forward into more advanced areas until you are able to understand 3D curves and surface patches, and solve problems using vectors.After reading the book, you should be able to refer to more challenging books with confidence and develop a greater insight into the design of computer graphics software.Get to grips with mathematics fast ...- Numbers- Algebra- Trigonometry- Coordinate geometry- Transforms- Vectors- Curves and surfaces- Analytic geometryEssential Mathematics for Computer Graphics fastThe book you will read once, and refer to over and over again!

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### Contents

 Is mathematics difficult? 3 Who should read this book? 4 Assumptions made in this book 5 Natural numbers 8 Integers 9 Irrational numbers 10 Summary 13 Algebra 15
 Rotating about an axis 96 3D reflections 98 2D change of axes 99 Direction cosines 100 Positioning the virtual camera 102 Direction cosines 103 Euler angles 106 Quaternions 110

 Notation 16 Algebraic laws 17 Associative law 18 Distributive law 19 Indices 20 Logarithms 21 Further notation 23 Trigonometry 25 The trigonometric ratios 26 Example 28 Trigonometric relationships 29 The cosine rule 30 Perimeter relationships 31 Summary 32 Cartesian Coordinates 33 The Cartesian xyplane 34 Function graphs 36 Geometric shapes 37 Areas of shapes 38 Theorem of Pythagoras in 2D 39 3D coordinates 40 3D polygons 41 Summary 42 Vectors 43 2D vectors 45 Magnitude of a vector 47 3D vectors 49 Multiplying a vector by a scalar 50 Position vectors 51 Unit vectors 52 Cartesian vectors 53 Vector multiplication 54 Scalar product 55 Example of the dot product 57 The dot product in lighting calculations 58 The dot product in backface detection 59 The vector product 60 The righthand rule 64 Deriving a unit normal vector for a triangle 65 Areas 66 Calculating 2D areas 67 Summary 68 Transformations 69 2D transformations 70 Reflection 71 Matrices 72 Systems of notation 75 The determinant of a matrix 76 Homogeneous coordinates 77 2D translation 79 2D reflections 80 2D shearing 82 2D rotation 83 2D scaling 86 2D reflections 87 2D rotation about an arbitrary point 88 3D transformations 89 3D scaling 90 Gimbal lock 95
 Multiplying quaternions 111 Rotating points about an axis 112 Roll pitch and yaw quaternions 116 Quaternions in matrix form 118 Frames of reference 119 Transforming vectors 120 Determinants 122 Perspective projection 126 Summary 128 Interpolation 129 Linear interpolant 130 Nonlinear interpolation 133 Cubic interpolation 135 Interpolating vectors 141 Interpolating quaternions 145 Summary 147 The circle 150 The ellipse 151 Bezier curves 152 Quadratic Bezier curves 157 Cubic Bernstein polynomials 158 A recursive Bezier formula 162 Linear interpolation 164 Bsplines 167 Uniform Bsplines 168 Continuity 171 Nonuniform Bsplines 172 Surface patches 173 Quadratic Bezier surface patch 174 Cubic Bezier surface patch 177 Summary 180 Review of geometry 182 Intercept theorems 183 Golden section 184 Triangles 185 Isosceles triangle 186 Equilateral triangle 187 Theorem of Pythagoras 188 Trapezoid 189 Rhombus 190 Circle 192 2D analytical geometry 193 The Hessian normal form 194 Space partitioning 197 The Hessian normal form from two points 198 Intersection points 199 Point inside a triangle 202 Hessian normal form 205 Intersection of a circle with a straight line 207 3D geometry 209 Point of intersection of two straight lines 211 Equation of a plane 214 Space partitioning 216 Point of intersection of a line segment and a plane 218 Summary 219 Conclusion 221 References 223 Copyright