Transformations and Projections in Computer Graphics
Transformations and projections are used extensively in Computer Graphics, a field which is now a part of everyone’s lives via feature films, advertisements in the media, the screens of PDAs, mobile phones, and other vehicles and outlets. Transformations
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David Salomon
Transformations and Projections in Computer Graphics
Professor David Salomon (emeritus) Computer Science Department California State University Northridge, CA 91330-8281 USA Email: [email protected]
Cover illustration: Adapted from figure 4.25, courtesy of Ari Salomon. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2006923906 ISBN-10: 1-84628-392-2 ISBN-13: 978-1-84628-392-5 Printed on acid-free paper © Springer-Verlag London Limited 2006 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 licences 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. Printed in the United States of America 9 8 7 6 5 4 3 2 1 springer.com
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Dedicated to Dick Termes, whose work and talent have contributed much to the quality of this book.
If there’s a book you really want to read, but it hasn’t been written yet, then you must write it.
—Toni Morrison
Preface It is probably a coincidence that the three main terms discussed in this book, namely transformations, projections, and perspective, are ambiguous. Here is what the dictionary has to say about these terms. Transformation (a) The act or an instance of transforming. (b) The state of being transformed. A marked change, as in appearance or character, usually for the better. Mathematical transformation. (a) Replacing a variable in an expression by its value. (b) Mapping a mathematical space onto another or onto itself. In geometry. Moving, rotating, reflecting, or otherwise systematically deforming a geometric figure (discussed in this book). In linguistics. (a) A rule to convert a syntactic form into another. (b) A sentence or sentential form derived by such a rule; a transform. In genetics. (a) The change undergone by a cell upon infection by a cancer-causing virus. (b) The alteration of a bacterial cell caused by the transfer of DNA from another bacterial cell, especially a pathogen. Projection The act of projecting or the condition of being projected. (a) An object or part thereof that extends outward. (b) Spiky projections on top of a fence. (c) A projection of land along the coast. A pr
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