Mathematics for Computer Graphics (eBook)

Contents 6
Preface 12
1 Mathematics 14
1.1 Is Mathematics Difficult? 15
1.2 Who should Read this Book? 15
1.3 Aims and Objectives of this Book 16
1.4 Assumptions Made in this Book 16
1.5 How to Use the Book 16
2 Numbers 18
2.1 Natural Numbers 18
2.2 Prime Numbers 19
2.3 Integers 19
2.4 Rational Numbers 19
2.5 Irrational Numbers 19
2.6 Real Numbers 20
2.7 The Number Line 20
2.8 Complex Numbers 20
2.9 Summary 22
3 Algebra 23
3.1 Notation 23
3.2 Algebraic Laws 24
3.3 Solving the Roots of a Quadratic Equation 26
3.4 Indices 27
3.5 Logarithms 27
3.6 Further Notation 28
3.7 Summary 28
4 Trigonometry 29
4.1 The Trigonometric Ratios 30
4.2 Example 30
4.3 Inverse Trigonometric Ratios 31
4.4 Trigonometric Relationships 31
4.5 The Sine Rule 32
4.6 The Cosine Rule 32
4.7 Compound Angles 32
4.8 Perimeter Relationships 33
4.9 Summary 34
5 Cartesian Coordinates 35
5.1 The Cartesian xy-plane 35
5.2 3D Coordinates 40
5.3 Summary 41
6 Vectors 42
6.1 2D Vectors 43
6.2 3D Vectors 45
6.3 Deriving a Unit Normal Vector for a Triangle 58
6.4 Areas 59
6.5 Summary 60
7 Transformation 61
7.1 2D Transformations 61
7.2 Matrices 63
7.3 Homogeneous Coordinates 67
7.4 3D Transformations 76
7.5 Change of Axes 83
7.6 Direction Cosines 85
7.7 Rotating a Point about an Arbitrary Axis 93
7.8 Transforming Vectors 108
7.9 Determinants 109
7.10 Perspective Projection 113
7.11 Summary 115
8 Interpolation 116
8.1 Linear Interpolant 116
8.2 Non-Linear Interpolation 119
8.3 Interpolating Vectors 125
8.4 Interpolating Quaternions 128
8.5 Summary 130
9 Curves and Patches 131
9.1 The Circle 131
9.2 The Ellipse 132
9.3 Bézier Curves 133
9.4 A recursive Bézier Formula 141
9.5 Bézier Curves Using Matrices 141
9.6 B-Splines 145
9.7 Surface Patches 149
9.8 Summary 154
10 Analytic Geometry 155
10.1 Review of Geometry 155
10.2 2D Analytical Geometry 164
10.3 Intersection Points 169
10.4 Point Inside a Triangle 172
10.5 Intersection of a Circle with a Straight Line 176
10.6 3D Geometry 177
10.7 Equation of a Plane 181
10.8 Intersecting Planes 189
10.9 Summary 199
11 Barycentric Coordinates 200
11.1 Ceva’s Theorem 200
11.2 Ratios and Proportion 202
11.3 Mass Points 203
11.4 Linear Interpolation 209
11.5 Convex Hull Property 215
11.6 Areas 216
11.7 Volumes 224
11.8 Bézier Curves and Patches 227
11.9 Summary 228
12 Worked Examples 229
12.1 Calculate the Area of a Regular Polygon 229
12.2 Calculate the Area of any Polygon 230
12.3 Calculate the Dihedral Angle of a Dodecahedron 230
12.4 Vector Normal to a Triangle 232
12.5 Area of a Triangle using Vectors 233
12.6 General Form of the Line Equation from Two Points 233
12.7 Calculate the Angle between Two Straight Lines 234
12.8 Test If Three Points Lie On a Straight Line 235
12.9 Find the Position and Distance of the Nearest Point on a Line to a Point 236
12.10 Position of a Point Re.ected in a Line 238
12.11 Calculate the Intersection of a Line and a Sphere 240
12.12 Calculate if a Sphere Touches a Plane 244
12.13 Summary 245
13 Conclusion 246
References 247
Index 248

6 Vectors (p. 31)

Vectors are a relatively new arrival to the world of mathematics, dating only from the 19th century. They provide us with some elegant and powerful techniques for computing angles between lines and the orientation of surfaces. They also provide a coherent framework for computing the behaviour of dynamic objects in computer animation and illumination models in rendering. We often employ a single number to represent quantities that we use in our daily lives such as, height, age, shoe size, waist and chest measurements. The magnitude of this number depends on our age and whether we use metric or imperial units. Such quantities are called scalars. In computer graphics scalar quantities include colour, height, width, depth, brightness, number of frames, etc.

On the other hand, there are some things that require more than one number to represent them: wind, force, weight, velocity and sound are just a few examples. These cannot be represented accurately by a single number. For example, any sailor knows that wind has a magnitude and a direction. The force we use to lift an object also has a value and a direction. Similarly, the velocity of a moving object is measured in terms of its speed (e.g. miles per hour) and a direction such as north-west. Sound, too, has intensity and a direction. These quantities are called vectors. In computer graphics, vectors are generally made of two or three numbers, and this is the only type we will consider in this chapter.

Mathematicians such as Caspar Wessel (1745–1818), Jean Argand (1768– 1822) and John Warren (1796–1852) were simultaneously exploring complex numbers and their graphical representation. In 1837, Sir William Rowan Hamilton (1788–1856) made his breakthrough with quaternions. In 1853, Hamilton published his book Lectures on Quaternions in which he described terms such as vector, transvector and provector. Hamilton’s work was not widely accepted until 1881, when the American mathematician Josiah Gibbs (1839–1903) published his treatise Vector Analysis, describing modern vector analysis.