3D Math Primer for Graphics and Game Development, 2nd Edition.pdf

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3D Math Primer for Graphics and Game Development Second Edition

3D Math Primer for Graphics and Game Development Second Edition Fletcher Dunn Ian Parberry

A K Peters/CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC A K Peters/CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works International Standard Book Number-13: 978-1-4398-6981-9 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the A K Peters Web site at http://www.akpeters.com

To A’me —F.D. To Maggie in the hope that she continues her interest in math —I. P.

Contents Acknowledgments Introduction 1 Cartesian Coordinate Systems 1.1 1.2 1.3 1.4 1.5 1D Mathematics . . . . . . . . . . . . . . . . . . . . . . 2D Cartesian Space . . . . . . . . . . . . . . . . . . . . . 3D Cartesian Space . . . . . . . . . . . . . . . . . . . . . Odds and Ends . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Vectors 2.1 Mathematical Deﬁnition of Vector, and Other Boring Stuﬀ . . . . . . . . . . . . . . Geometric Deﬁnition of Vector 2.2 Specifying Vectors with Cartesian Coordinates . . . . . 2.3 Vectors versus Points . . . . . . . . . . . . . . . . . . . . 2.4 Negating a Vector . . . . . . . . . . . . . . . . . . . . . . 2.5 Vector Multiplication by a Scalar . . . . . . . . . . . . . 2.6 Vector Addition and Subtraction . . . . . . . . . . . . . 2.7 Vector Magnitude (Length) . . . . . . . . . . . . . . . . 2.8 Unit Vectors . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 2.10 The Distance Formula . . . . . . . . . . . . . . . . . . . 2.11 Vector Dot Product . . . . . . . . . . . . . . . . . . . . . 2.12 Vector Cross Product . . . . . . . . . . . . . . . . . . . . 2.13 Linear Algebra Identities . . . . . . . . . . . . . . . . . . 2.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Multiple Coordinate Spaces 3.1 Why Bother with Multiple Coordinate Spaces? . . . . . 3.2 Some Useful Coordinate Spaces . . . . . . . . . . . . . . Basis Vectors and Coordinate Space Transformations . . 3.3 3.4 Nested Coordinate Spaces . . . . . . . . . . . . . . . . . xiii xv 1 2 5 12 19 27 31 32 34 36 39 43 45 47 51 53 55 56 66 70 71 79 80 81 86 106 vii

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