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 f
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John Vince
Quaternions for Computer Graphics
Professor John Vince, MTech, PhD, DSc, CEng, FBCS Bournemouth University, Bournemouth, UK url: www.johnvince.co.uk
ISBN 978-0-85729-759-4 e-ISBN 978-0-85729-760-0 DOI 10.1007/978-0-85729-760-0 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: 2011931282 © Springer-Verlag London Limited 2011 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: VTeX UAB, Lithuania Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
This book is dedicated to Heidi
Preface
More than 50 years ago when I was studying to become an electrical engineer, I came across complex numbers, which were used to represent out-of-phase voltages and currents using the j operator. I believe that the letter j was used, rather than i, because the latter stood for electrical current. So from the very start of my studies I had a clear mental picture of the imaginary unit as a rotational operator which could advance or retard electrical quantities in time. When events dictated that I would pursue a career in computer programming— rather than electrical engineering—I had no need for complex numbers, until Mandlebrot’s work on fractals emerged. But that was a temporary phase, and I never needed to employ complex numbers in any of my computer graphics software. However in 1986, when I joined the flight simulation industry, I came across an internal report on quaternions, which were being used to control the rotational orientation of a simulated aircraft. I can still remember being completely bemused by quaternions, simply because they involved so many imaginary terms. However, after much research I started to understand what they were, but not how they worked. Simultaneously, I was becoming interested in the philosophical side of mathematics, and trying to come to terms with the ‘real meaning’ of mathematics through the writing of Bertrand Russell. Consequently, concepts such as i we
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