Essential Mathematics for Computer Graphics fast

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1 John Vince Essential Mathematics for Computer Graphics fast Springer

2 Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made in this book 5 How to use the book 5 2. NUMBERS 7 Natural numbers 8 Prime numbers 8 Integers 9 Rational numbers 9 Irrational numbers 10 Real numbers 10 The number line 10 Complex numbers 10 Summary ALGEBRA 15 Notation 16 Algebraic laws 17 Associative law 18 Commutative law 18 Distributive law 19 Solving the roots of a quadratic equation 19 Indices 20 Laws of indices 21 Examples 21 Logarithms 21 Further notation 23 Summary TRIGONOMETRY 25 The trigonometric ratios 26 Example 28 Inverse trigonometric ratios 28 Trigonometric relationships 29 The sine rule 29

3 Essential Mathematics for CG fast The cosine rule.'. 30 Compound angles 30 Perimeter relationships 31 Summary CARTESIAN COORDINATES 33 The Cartesian xy-plane 34 Function graphs 36 Geometric shapes 37 Polygonal shapes 37 Areas of shapes 38 Theorem of Pythagoras in 2D 39 3D coordinates 40 Theorem of Pythagoras in 3D 40 3D polygons 41 Euler's rule 41 Summary VECTORS 43 2D vectors 45 Vector notation 45 Graphical representation of vectors 45 Magnitude of a vector 47 3D vectors 49 Vector manipulation 49 Multiplying a vector by a scalar 50 Vector addition and subtraction 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 back-face detection 59 The vector product 60 The right-hand rule 64 Deriving a unit normal vector for a triangle 65 Areas 66 Calculating 2D areas 67 Summary 68

4 Contents 7. TRANSFORMATIONS 69 2D transformations 70 Translation 70 Scaling 70 Reflection 71 Matrices 72 Systems of notation 75 The determinant of a matrix 76 Homogeneous coordinates 77 2D translation 79 2D scaling 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 translation 89 3D scaling 90 3D rotations 90 Gimbal lock 95 Rotating about an axis 96 3D reflections 98 Change of axes 98 2D change of axes 99 Direction cosines 100 Positioning the virtual camera 102 Direction cosines 103 Euler angles 106 Quaternions 110 Adding and subtracting quaternions 110 Multiplying quaternions 111 The inverse quaternion 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

5 Essential Mathematics for CG fast 8. INTERPOLATION 129 Linear interpolant 130 Non-linear interpolation 133 Trigonometric interpolation 133 Cubic interpolation 135 Interpolating vectors 141 Interpolating quaternions 145 Summary CURVES AND PATCHES 149 The circle 150 The ellipse 151 Bezier curves 152 Bernstein polynomials 152 Quadratic Bezier curves 157 Cubic Bernstein polynomials 158 A recursive Bezier formula 162 Bezier curves using matrices 162 Linear interpolation 164 B-splines 167 Uniform B-splines 168 Continuity 171 Non-uniform B-splines 172 Non-uniform Rational B-Splines 172 Surface patches 173 Planar surface patch 173 Quadratic Bezier surface patch 174 Cubic Bezier surface patch 177 Summary ANALYTIC GEOMETRY 181 Review of geometry 182 Angles 182 Intercept theorems 183 Golden Section 184 Triangles 185 Center of gravity of a triangle 185 Isosceles triangle 186 Equilateral triangle 187 Right triangle 187 Theorem of Thales 187 Theorem of Pythagoras 188 Quadrilateral 188

6 Contents Trapezoid 189 Parallelogram 189 Rhombus 190 Regular polygon (n-gon) 190 Circle 192 2D analytic geometry 193 Equation of a straight line 193 The Hessian normal form 194 Space partitioning 197 The Hessian normal form from two points 198 Intersection points 199 Intersection point of two straight lines 199 Intersection point of two line segments 199 Point inside a triangle 202 Area of a triangle 202 Hessian normal form 205 Intersection of a circle with a straight line 207 3D geometry 209 Equation of a straight line 209 Point of intersection of two straight lines 211 Equation of a plane 215 Space partitioning 216 Point of intersection of a line and a plane 216 Point of intersection of a line segment and a plane 218 Summary CONCLUSION 221 REFERENCES 223 INDEX 225

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