Preface 6
Contents 8
1 Mathematics 15
1.1 Introduction 15
1.2 Is Mathematics Difficult? 16
1.3 Who Should Read This Book? 16
1.4 Aims and Objectives of This Book 17
1.5 Assumptions Made in This Book 17
1.6 How to Use This Book 17
2 Numbers 18
2.1 Introduction 18
2.2 Natural Numbers 19
2.3 Prime Numbers 19
2.4 Integers 19
2.5 Rational Numbers 20
2.6 Irrational Numbers 20
2.7 Real Numbers 20
2.8 The Number Line 20
2.9 Complex Numbers 21
2.10 Summary 24
3 Algebra 25
3.1 Introduction 25
3.2 Notation 25
3.3 Algebraic Laws 26
3.3.1 Associative Law 27
3.3.2 Commutative Law 27
3.3.3 Distributive Law 28
3.4 Solving the Roots of a Quadratic Equation 28
3.5 Indices 29
3.5.1 Laws of Indices 29
3.5.2 Examples 30
3.6 Logarithms 30
3.7 Further Notation 31
3.8 Summary 31
4 Trigonometry 32
4.1 Introduction 32
4.2 The Trigonometric Ratios 33
4.3 Example 34
4.4 Inverse Trigonometric Ratios 34
4.5 Trigonometric Relationships 34
4.6 The Sine Rule 35
4.7 The Cosine Rule 35
4.8 Compound Angles 36
4.9 Perimeter Relationships 36
4.10 Summary 37
5 Cartesian Coordinates 38
5.1 Introduction 38
5.2 The Cartesian xy-Plane 38
5.2.1 Function Graphs 39
5.2.2 Geometric Shapes 40
5.2.3 Polygonal Shapes 40
5.2.4 Areas of Shapes 41
5.2.5 Theorem of Pythagoras in 2D 42
5.3 3D Coordinates 42
5.3.1 Theorem of Pythagoras in 3D 44
5.3.2 3D Polygons 44
5.3.3 Euler's Rule 44
5.4 Summary 44
6 Vectors 46
6.1 Introduction 46
6.2 2D Vectors 47
6.2.1 Vector Notation 47
6.2.2 Graphical Representation of Vectors 48
6.2.3 Magnitude of a Vector 49
6.3 3D Vectors 50
6.3.1 Vector Manipulation 51
6.3.2 Multiplying a Vector by a Scalar 51
6.3.3 Vector Addition and Subtraction 52
6.3.4 Position Vectors 53
6.3.5 Unit Vectors 54
6.3.6 Cartesian Vectors 55
6.3.7 Vector Multiplication 56
6.3.8 Scalar Product 56
6.3.9 Example of the Scalar Product 58
6.3.10 The Dot Product in Lighting Calculations 59
6.3.11 The Scalar Product in Back-Face Detection 60
6.3.12 The Vector Product 61
6.3.13 The Right-Hand Rule 65
6.4 Deriving a Unit Normal Vector for a Triangle 66
6.5 Areas 66
6.5.1 Calculating 2D Areas 67
6.6 Summary 68
7 Transforms 69
7.1 Introduction 69
7.2 2D Transforms 69
7.2.1 Translation 69
7.2.2 Scaling 70
7.2.3 Reflection 70
7.3 Matrices 72
7.3.1 Systems of Notation 74
7.3.2 The Determinant of a Matrix 75
7.4 Homogeneous Coordinates 75
7.4.1 2D Translation 77
7.4.2 2D Scaling 77
7.4.3 2D Reflections 78
7.4.4 2D Shearing 80
7.4.5 2D Rotation 80
7.4.6 2D Scaling 83
7.4.7 2D Reflection 83
7.4.8 2D Rotation About an Arbitrary Point 84
7.5 3D Transforms 85
7.5.1 3D Translation 85
7.5.2 3D Scaling 86
7.5.3 3D Rotation 86
7.5.4 Gimbal Lock 90
7.5.5 Rotating About an Axis 91
7.5.6 3D Reflections 93
7.6 Change of Axes 93
7.6.1 2D Change of Axes 93
7.6.2 Direction Cosines 95
7.6.3 3D Change of Axes 96
7.7 Positioning the Virtual Camera 97
7.7.1 Direction Cosines 97
7.7.2 Euler Angles 100
7.8 Rotating a Point About an Arbitrary Axis 103
7.8.1 Matrices 103
7.8.2 Quaternions 109
7.8.3 Adding and Subtracting Quaternions 110
7.8.4 Multiplying Quaternions 111
7.8.5 Pure Quaternion 111
7.8.6 The Inverse Quaternion 112
7.8.7 Unit Quaternion 112
7.8.8 Rotating Points About an Axis 112
7.8.9 Roll, Pitch and Yaw Quaternions 116
7.8.10 Quaternions in Matrix Form 117
7.8.11 Frames of Reference 118
7.9 Transforming Vectors 119
7.10 Determinants 121
7.11 Perspective Projection 125
7.12 Summary 127
8 Interpolation 128
8.1 Introduction 128
8.2 Linear Interpolation 128
8.3 Non-Linear Interpolation 131
8.3.1 Trigonometric Interpolation 131
8.3.2 Cubic Interpolation 132
8.4 Interpolating Vectors 137
8.5 Interpolating Quaternions 140
8.6 Summary 142
9 Curves and Patches 143
9.1 Introduction 143
9.2 The Circle 143
9.3 The Ellipse 144
9.4 B茅zier Curves 145
9.4.1 Bernstein Polynomials 145
9.4.2 Quadratic B茅zier Curves 149
9.4.3 Cubic Bernstein Polynomials 150
9.5 A Recursive B茅zier Formula 153
9.6 B茅zier Curves Using Matrices 153
9.6.1 Linear Interpolation 154
9.7 B-Splines 157
9.7.1 Uniform B-Splines 158
9.7.2 Continuity 160
9.7.3 Non-uniform B-Splines 161
9.7.4 Non-uniform Rational B-Splines 161
9.8 Surface Patches 162
9.8.1 Planar Surface Patch 162
9.8.2 Quadratic B茅zier Surface Patch 163
9.8.3 Cubic B茅zier Surface Patch 165
9.9 Summary 168
10 Analytic Geometry 169
10.1 Introduction 169
10.2 Review of Geometry 169
10.2.1 Angles 169
10.2.2 Intercept Theorems 170
10.2.3 Golden Section 171
10.2.4 Triangles 171
10.2.5 Centre of Gravity of a Triangle 172
10.2.6 Isosceles Triangle 172
10.2.7 Equilateral Triangle 173
10.2.8 Right Triangle 173
10.2.9 Theorem of Thales 173
10.2.10 Theorem of Pythagoras 174
10.2.11 Quadrilaterals 174
10.2.12 Trapezoid 175
10.2.13 Parallelogram 175
10.2.14 Rhombus 176
10.2.15 Regular Polygon (n-gon) 176
10.2.16 Circle 176
10.3 2D Analytic Geometry 178
10.3.1 Equation of a Straight Line 178
10.3.2 The Hessian Normal Form 179
10.3.3 Space Partitioning 181
10.3.4 The Hessian Normal Form from Two Points 182
10.4 Intersection Points 183
10.4.1 Intersection Point of Two Straight Lines 183
10.4.2 Intersection Point of Two Line Segments 183
10.5 Point Inside a Triangle 186
10.5.1 Area of a Triangle 186
10.5.2 Hessian Normal Form 188
10.6 Intersection of a Circle with a Straight Line 190
10.7 3D Geometry 191
10.7.1 Equation of a Straight Line 191
10.7.2 Point of Intersection of Two Straight Lines 193
10.8 Equation of a Plane 195
10.8.1 Cartesian Form of the Plane Equation 196
10.8.2 General Form of the Plane Equation 198
10.8.3 Parametric Form of the Plane Equation 198
10.8.4 Converting from the Parametric to the General Form 200
10.8.5 Plane Equation from Three Points 201
10.9 Intersecting Planes 203
10.9.1 Intersection of Three Planes 207
10.9.2 Angle Between Two Planes 209
10.9.3 Angle between a Line and a Plane 211
10.9.4 Intersection of a Line with a Plane 212
10.10 Summary 214
11 Barycentric Coordinates 215
11.1 Introduction 215
11.2 Ceva's Theorem 215
11.3 Ratios and Proportion 217
11.4 Mass Points 218
11.5 Linear Interpolation 224
11.6 Convex Hull Property 231
11.7 Areas 231
11.8 Volumes 240
11.9 B茅zier Curves and Patches 242
11.10 Summary 243
12 Geometric Algebra 244
12.1 Introduction 244
12.2 Symmetric and Antisymmetric Functions 244
12.3 Trigonometric Foundations 246
12.4 Vectorial Foundations 248
12.5 Inner and Outer Products 248
12.6 The Geometric Product in 2D 249
12.7 The Geometric Product in 3D 252
12.8 The Outer Product of Three 3D Vectors 253
12.9 Axioms 255
12.10 Notation 255
12.11 Grades, Pseudoscalars and Multivectors 256
12.12 Redefining the Inner and Outer Products 257
12.13 The Inverse of a Vector 257
12.14 The Imaginary Properties of the Outer Product 259
12.15 Duality 261
12.16 The Relationship Between the Vector Product and the Outer Product 262
12.17 The Relationship Between Quaternions and Bivectors 262
12.18 Reflections and Rotations 263
12.18.1 2D Reflections 264
12.18.2 3D Reflections 264
12.18.3 2D Rotations 265
12.19 Rotors 267
12.20 Applied Geometric Algebra 271
12.20.1 Sine Rule 271
12.20.2 Cosine Rule 272
12.20.3 A Point Perpendicular to a Point on a Line 272
12.20.4 Reflecting a Vector about a Vector 274
12.20.5 Orientation of a Point with a Plane 275
12.21 Summary 277
13 Worked Examples 278
13.1 Introduction 278
13.2 Area of Regular Polygon 278
13.3 Area of any Polygon 279
13.4 Dihedral Angle of a Dodecahedron 280
13.5 Vector Normal to a Triangle 281
13.6 Area of a Triangle Using Vectors 282
13.7 General Form of the Line Equation from Two Points 282
13.8 Angle Between Two Straight Lines 283
13.9 Test if Three Points Lie on a Straight Line 284
13.10 Position and Distance of the Nearest Point on a Line to a Point 285
13.11 Position of a Point Reflected in a Line 287
13.12 Intersection of a Line and a Sphere 289
13.13 Sphere Touching a Plane 293
13.14 Summary 295
14 Conclusion 296
Index 297