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Geometry for Computer Graphics

John Vince

Geometry for

Computer Graphics

Formulae, Examples and Proofs

123

John Vince MTech, PhD, CEng, FBCS

National Centre for Computer Animation

Bournemouth University, UK

British Library Cataloguing in Publication Data

Vince, John (John A.)

Geometry for computer graphics : formulae, examples and proofs

1. Computer graphics 2. Geometry – Data processing

I. Title

516

Library of Congress Cataloging-in-Publication Data

A catalog record for this book is available from the Library of Congress

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the

or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in

accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction

outside those terms should be sent to the publishers.

ISBN 1-85233-834-2 Springer-Verlag London Berlin Heidelberg

Springer Science

Printed in the United States of America

The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a speciﬁc

statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.

The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in

this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made.

Typesetting: Gray Publishing, Tunbridge Wells, UK

34/3830-543210 Printed on acid-free paper SPIN 10981696

.0028566

ISBN 1852338342

′

Business Media

springeronline.com

Dedication

This book is dedicated to my family, Annie, Samantha, Anthony, Megan, and Monty, who have

not seen much of me over the past two years.

Preface

Anyone who has written programs for computer graphics, CAD, scientiﬁc visualization, computer

games, virtual reality or computer animation will know that mathematics is extremely useful.

Topics such as transformations, matrix algebra, vector algebra, curves and surfaces are at the

heart of any application program in these areas, but the one topic that is really central is geometry,

which is the theme of this book.

I recall many times when writing computer animation programs my own limited knowledge

of geometry. I remember once having to create a 3D lattice of dodecahedrons as the basis for a

cell growth model. At the time, I couldn’t ﬁnd a book on the subject and had to compute Platonic

solid dihedral angles and vertex coordinates from scratch. The Internet had not been invented

and I was left to my own devices to solve the problem. As it happened, I did solve it, and my new

found knowledge of Platonic objects has never waned.

Fortunately, I no longer have to write computer programs, but many other people still do, and

the need for geometry has not gone away. In fact, as computer performance has increased, it has

become possible to solve amazingly complex three-dimensional geometric problems in real time.

The reason for writing this book is threefold: to begin with, I wanted to coordinate a wealth of

geometry that is spread across all sorts of math books and the Internet; second, I wanted to illustrate

how a formula was used in practice; third, I wanted to provide simple proofs for these formulas.

Personally, whenever I see an equation I want to know its origin. For example, why is the volume

of a tetrahedron one-sixth of a set of vertices? Where does the ‘one-sixth’ come from? Take another

example: why is the volume of a sphere four-thirds, p, radius cubed? Where does the ‘four-thirds’

come from? Why isn’t it ‘ﬁve-sixths’? This may be a personal problem I have about the origins of

formulas but I do ﬁnd that my understanding of a subject is increased when I understand its origins.

Quaternions are another example. There is still some mystique about what they are and how

they work. I can think of no better way of understanding quaternions than to read about Sir

William Rowan Hamilton and discover how he stumbled across his now famous non-commutative

algebra.

I am the ﬁrst to admit that I am not a mathematician, and this book is not intended to be read

by mathematicians. A mathematician would have approached the subject with a greater logical

rigour and employed formal structures that are relevant to the world of mathematics, but of

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