# Mathematics for Computer Graphics (4th Edition)

## by John Vince

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This book is not for mathematicians! They would probably raise their hands in horror about the level of mathematical rigour I have employed, or probably not employed! This book is for people working in computer graphics who know that they have to use mathematics in their day-to-day work, and don't want to get too embroiled in axioms, truths, and Platonic realities.

This book originally appeared as part of Springer's excellent *Essential* series, and was revised to include chapters on analytical geometry, barycentric coordinates, and worked examples. The third edition included a new chapter on geometric algebra, which I have written about in my books *Geometric Algebra for Computer Graphics* and *Geometric Algebra: An Algebraic System for Computer Games and Animation*.In this fourth edition, I have reviewed the entire book and included chapters on differential and integral calculus, which I have written about in *Calculus for Computer Graphics*.

Whilst writing this book I have borne in mind what it was like for me when I was studying different areas of mathematics for the first time. In spite of reading and rereading an explanation several times it could take days before "the penny dropped" and a concept became apparent. Hopefully, the reader will find the explanations useful in developing their understanding of these specific areas of mathematics, and enjoy the sound of various pennies dropping!

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Calculus is one of those subjects that appears to have no boundaries, which is why some calculus books are so large and heavy! So when I started writing this book, I knew that it would not fall into this category: it would be around 200 pages long and take the reader on a gentle journey through the subject, without placing too many demands on their knowledge of mathematics.

The objective of the book is to inform the reader about functions and their derivatives, and the inverse process: integration, which can be used for computing area and volume. The emphasis on geometry gives the book relevance to the computer graphics community, and hopefully will provide the mathematical background for professionals working in computer animation, games and allied disciplines to read and understand other books and technical papers where differential and integral notation is found.

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Matrix transforms are ubiquitous within the world of computer graphics, where they have become an invaluable tool in a programmer’s toolkit for solving everything from 2D image scaling to 3D rotation about an arbitrary axis. Virtually every software system and hardware graphics processor uses matrices to undertake operations such as scaling, translation, reflection and rotation. Nevertheless, for come newcomers to the world of computer games and animation, matrix notation can appear obscure and challenging.

Matrices and determinants were originally used to solve groups of simultaneous linear equations, and were subsequently embraced by the computer graphics community to describe the geometric operations for manipulating two- and three-dimensional structures. Consequently, to place matrix notation within an historical context, I provide readers with some useful background to their development, alongside determinants.

Although this book assumes that the reader is familiar with everyday algebra and the solution of simultaneous linear equations, it does not expect any prior knowledge of matrix notation. The book includes chapters on matrix notation, determinants, matrices, 2D transforms, 3D transforms and quaternions, and includes many worked examples to illustrate their practical use. After reading this book, readers should be completely familiar with matrix notation and their application within the world of computer games and animation.

# Quaternions for Computer Graphics

Sir William Rowan Hamilton
was a genius, and will be remembered for his significant contributions to
physics and mathematics. The *Hamiltonian,*
which is used in quantum physics to describe the total energy of a system,
would have been a major achievement for anyone, but Hamilton also invented
quaternions, which paved the way for modern vector analysis.

Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive.

*Quaternions for Computer Graphics* introduces the reader to quaternion algebra by
describing concepts of sets, groups, fields and rings. It also includes
chapters on imaginary quantities, complex numbers and the complex plane, which
are essential to understanding quaternions. The book contains many
illustrations and worked examples, which make it essential reading for
students, academics, researchers and professional practitioners.

# Rotation Transforms for Computer Graphics

by John Vince

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Rotation transforms are used everywhere in computer graphics from rotating pictures in editing software, to providing an arbitrary view of a 3D virtual environment. Although the former is a trivial operation, the latter can be a challenging task.

*Rotation Transforms for Computer Graphics* covers a wide range of mathematical techniques used
for rotating points and frames of reference in the plane and 3D space. It
includes many worked examples and over 100 illustrations that make it essential
reading for students, academics, researchers and professional practitioners.The book includes
introductory chapters on complex numbers, matrices, quaternions and geometric
algebra, and further chapters on how these techniques are employed in 2D and 3D
computer graphics. In particular, matrix and bivector transforms are developed
and evaluated to rotate points in a fixed frame of reference, and vice versa.

# Mathematics for Computer Graphics (3rd Edition)

## by John Vince

## BUY ON AMAZON

This book is not for mathematicians! They would probably raise their hands in horror about the level of mathematical rigour I have employed, or probably not employed! This book is for people working in computer graphics who know that they have to use mathematics in their day-to-day work, and don't want to get too embroiled in axioms, truths, and Platonic realities.

This book originally appeared as part of Springer's excellent *Essential* series, and was revised to include chapters on analytical geometry, barycentric coordinates, and worked examples. The third edition included a new chapter on geometric algebra, which I have written about in my books *Geometric Algebra for Computer Graphics* and *Geometric Algebra: An Algebraic System for Computer Games and Animation*.

Whilst writing this book I have borne in mind what it was like for me when I was studying different areas of mathematics for the first time. In spite of reading and rereading an explanation several times it could take days before "the penny dropped" and a concept became apparent. Hopefully, the reader will find the explanations useful in developing their understanding of these specific areas of mathematics, and enjoy the sound of various pennies dropping!

# Geometric Algebra: An Algebraic System for Computer Games and Animation

## by John Vince

*geometric algebra*. As such geometric elements are central to the world of computer games and computer animation, geometric algebra offers programmers new ways of solving old problems.

Filled with lots of clear examples, full-colour illustrations and tables, this compact book provides an excellent introduction to geometric algebra for practitioners in computer games and animation.

Geometric Algebra for Computer Graphics

## by John Vince

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Since its invention, geometric algebra has been applied to various branches of physics such as cosmology and electrodynamics, and is now being embraced by the computer graphics community where it is providing new ways of solving geometric problems. It took over two-thousand years to discover this algebra, which uses a simple and consistent notation to describe vectors and their products.

The first five chapters review the algebras of real numbers, complex numbers, vectors, and quaternions and their associated axioms, together with the geometric conventions employed in analytical geometry. As well as putting geometric algebra into its historical context, I provide chapters on Grassmann's outer product and Clifford's geometric product, followed by the application of geometric algebra to reflections, rotations, lines, planes and their intersection. The conformal model is also covered, where a 5D Minkowski space provides an unusual platform for unifying the transforms associated with 3D Euclidean space.

Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to geometric algebra for computer graphics.

# Vector Analysis for Computer Graphics

## by John Vince

Vector analysis is relatively young in the history of mathematics, however, in the short period of its existence it has become a powerful and central tool in describing and solving a wide range of geometric problems, many, of which, arise in computer graphics. These may be in the form of describing lines, surfaces and volumes, which may touch, collide, intersect, or create shadows upon complex surfaces.

*Vector
Analysis for Computer Graphics* provides a complete
introduction to vector analysis, especially within the context of computer
graphics. The author shows why vectors are useful and how it is possible to
develop analytical skills in manipulating the vector algebra. Each topic
covered is placed in the context of a practical application within computer
graphics.

The book is divided into eleven chapters covering the mathematical foundations of vector algebra and its application to lines, planes, intersections, rotating vectors, vector differentiation, projections, rendering and motion.

# Mathematics for Computer Graphics (2nd Edition)

## by John Vince

# Geometry for Computer Graphics

## by John Vince

Geometry is the cornerstone of computer graphics and computer animation, and provides the framework and tools for solving problems in two and three dimensions. This may be in the form of describing simple shapes such as a circle, ellipse or parabola, or complex problems such as rotating objects about an arbitrary axis.

Geometry for Computer Graphics draws together a wide variety of geometric information that will provide a sourcebook of facts, examples and proofs for students, academics, researchers and professional practitioners.

# Introduction to Virtual Reality

## by John Vince

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## by John Vince

## by John Vince

I tell you about the principles used in the powerful software products currently available on the market; the terms and processes involved; and in an easy-to-understand way, with no complicated math.

So if you want to learn more about 3D computer animation without being swamped by complex mathematics, then read this book and have fun creating your own animated programs.

## by John Vince

# 3-D Computer Animation

## by John Vince

3D Computer Animation presents a clear and accessible guide to modern techniques for three-dimensional computer animation. Richly illustrated, it describes the major techniques and algorithms used to produce animated computer graphics in television, advertising, film special effects and flight simulators.

# 3-D Computer Animation

## by John Vince

3D Computer Animation presents a clear and accessible guide to modern techniques for three-dimensional computer animation. Richly illustrated, it describes the major techniques and algorithms used to produce animated computer graphics in television, advertising, film special effects and flight simulators.