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Mathematics for Computer Graphics

:	John Vince
:	Mathematics for Computer Graphics
:	Springer-Verlag
:	9781846282836
:	2
:	CHF 32.90
:

:	Informatik
:	English

:	251
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

This is a concise and informal introductory book on the mathematical concepts that underpin computer graphics. The author, John Vince, makes the concepts easy to understand, enabling non-experts to come to terms with computer animation work. The book complements the author's other works and is written in the same accessible and easy-to-read style. It is also a useful reference book for programmers working in the field of computer graphics, virtual reality, computer animation, as well as students on digital media courses, and even mathematics courses.

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.

	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