Mathematics for Computer Graphics

John Vince explains a wide range of mathematical techniques and problem-solving strategies associated with computer games, computer animation, virtual reality, CAD, and other areas of computer graphics. Covering all the mathematical techniques required to

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Undergraduate Topics in Computer Science’ (UTiCS) delivers high-quality instructional content for undergraduates studying in all areas of computing and information science. From core foundational and theoretical material to final-year topics and applications, UTiCS books take a fresh, concise, and modern approach and are ideal for self-study or for a one- or two-semester course. The texts are all authored by established experts in their fields, reviewed by an international advisory board, and contain numerous examples and problems. Many include fully worked solutions.

For other volumes: http://www.springer.com/series/7592

John Vince

Mathematics for Computer Graphics 3rd Edition

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Prof. John Vince, MTech, PhD, DSc, CEng, FBCS www.johnvince.co.uk Series editor Ian Mackie

Advisory board Samson Abramsky, University of Oxford, UK Chris Hankin, Imperial College London, UK Dexter Kozen, Cornell University, USA Andrew Pitts, University of Cambridge, UK Hanne Riis Nielson, Technical University of Denmark, Denmark Steven Skiena, Stony Brook University, USA Iain Stewart, University of Durham, UK David Zhang, The Hong Kong Polytechnic University, Hong Kong

Undergraduate Topics in Computer Science ISSN 1863-7310 ISBN: 978-1-84996-022-9 e-ISBN: 978-1-84996-023-6 DOI: 10.1007/978-1-84996-023-6 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942716 c Springer-Verlag London Limited 2010  Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form 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 licenses issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. 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. Cover design: SPi Publisher Services Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

This book is dedicated to my grandchildren: Megan, Mia and Lucie.

Preface

Mathematics is a beautiful subject. Its symbols, notation and abstract structures permit us to define, manipulate and resolve extremely complex problems. However, the symbols by themselves are meaningless – they are nothing more than a calligraphic representation of a mental idea. If one does not understan

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