Rotation Transforms For Computer Graphics

Rotation Transforms for Computer Graphics John Vince Rotation Transforms for Computer Graphics Professor Emeritus, John Vince, MTech, PhD, DSc, CEng, FBCS Bournemouth University, Bournemouth, UK url: www.johnvince.co.uk ISBN 978-0-85729-153-0 e-ISBN 978-0-85729-154-7 DOI 10.1007/978-0-85729-154-7 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library © 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: deblik 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 Every time I complete a manuscript my attention turns quickly to the title of my next book. And after completing the latest version of Mathematics for Computer Graphics, I began to think of what should follow. It didn’t take too long to identify the subject of this book: rotation transforms, which have always interested me throughout my career in computer graphics. I knew that I was not alone in finding some of the ideas difficult, as every time I searched the Internet using search keys such as ‘quaternions’, ‘Euler angles’, ‘rotors’, etc., I would come across websites where groups were discussing the meaning of gimbal lock, the matrix representation of a quaternion, eigenvectors, etc., and I knew straight away that I had to do my bit to clarify the subject. One of the main problems why there is so much confusion arises through the different forms of vector and matrix notation. Some authors work with matrices that involve row vectors, rather than column vectors, which leads to a transposed matrix. In some cases, the direction of rotation is clockwise, rather than the normal positive, anti-clockwise direction. Quaternions are treated as a four-dimensional object where one has to visuali