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Quaternions for Computer Graphics by John Vince.pdf

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    Cover
    Quaternions for Computer Graphics
    ISBN 9780857297594
    Preface
    Contents
    Chapter 1: Introduction
    1.1 Rotation Transforms
    1.2 The Reader
    1.3 Aims and Objectives of This Book
    1.4 Mathematical Techniques
    1.5 Assumptions Made in This Book
    Chapter 2: Number Sets and Algebra
    2.1 Introduction
    2.2 Number Sets
    2.2.1 Natural Numbers
    2.2.2 Real Numbers
    2.2.3 Integers
    2.2.4 Rational Numbers
    2.3 Arithmetic Operations
    2.4 Axioms
    2.5 Expressions
    2.6 Equations
    2.7 Ordered Pairs
    2.8 Groups, Rings and Fields
    2.8.1 Groups
    2.8.2 Abelian Group
    2.8.3 Rings
    2.8.4 Fields
    2.8.5 Division Ring
    2.9 Summary
    2.9.1 Summary of Definitions
    Chapter 3: Complex Numbers
    3.1 Introduction
    3.2 Imaginary Numbers
    3.3 Powers of i
    3.4 Complex Numbers
    3.5 Adding and Subtracting Complex Numbers
    3.6 Multiplying a Complex Number by a Scalar
    3.7 Complex Number Products
    3.7.1 Square of a Complex Number
    3.8 Norm of a Complex Number
    3.9 Complex Conjugate
    3.10 Quotient of Two Complex Numbers
    3.11 Inverse of a Complex Number
    3.12 Square-Root of i
    3.13 Field Structure
    3.14 Ordered Pairs
    3.14.1 Multiplying by a Scalar
    3.14.2 Complex Conjugate
    3.14.3 Quotient
    3.14.4 Inverse
    3.15 Matrix Representation of a Complex Number
    3.15.1 Adding and Subtracting
    3.15.2 The Product
    3.15.3 The Square of the Norm
    3.15.4 The Complex Conjugate
    3.15.5 The Inverse
    3.15.6 Quotient
    3.16 Summary
    3.16.1 Summary of Operations
    3.17 Worked Examples
    Chapter 4: The Complex Plane
    4.1 Introduction
    4.2 Some History
    4.3 The Complex Plane
    4.4 Polar Representation
    4.5 Rotors
    4.6 Summary
    4.6.1 Summary of Operations
    4.7 Worked Examples
    Chapter 5: Quaternion Algebra
    5.1 Introduction
    5.2 Some History
    5.3 Defining a Quaternion
    5.3.1 The Quaternion Units
    5.3.2 Example of Quaternion Products
    5.4 Algebraic Definition
    5.5 Adding and Subtracting Quaternions
    5.6 Real Quaternion
    5.7 Multiplying a Quaternion by a Scalar
    5.8 Pure Quaternion
    5.9 Unit Quaternion
    5.10 Additive Form of a Quaternion
    5.11 Binary Form of a Quaternion
    5.12 The Conjugate
    5.13 Norm of a Quaternion
    5.14 Normalised Quaternion
    5.15 Quaternion Products
    5.15.1 Product of Pure Quaternions
    5.15.2 Product of Two Unit-Norm Quaternions
    5.15.3 Square of a Quaternion
    5.15.4 Norm of the Quaternion Product
    5.16 Inverse Quaternion
    5.17 Matrices
    5.17.1 Orthogonal Matrix
    5.18 Quaternion Algebra
    5.19 Summary
    5.19.1 Summary of Operations
    5.20 Worked Examples
    Chapter 6: 3D Rotation Transforms
    6.1 Introduction
    6.2 3D Rotation Transforms
    6.3 Rotating About a Cartesian Axis
    6.4 Rotate About an Off-Set Axis
    6.5 Composite Rotations
    6.6 Rotating About an Arbitrary Axis
    6.6.1 Matrices
    6.6.2 Vectors
    6.7 Summary
    6.7.1 Summary of Transforms
    6.8 Worked Examples
    Chapter 7: Quaternions in Space
    7.1 Introduction
    7.2 Some History
    7.2.1 Composition Algebras
    7.3 Quaternion Products
    7.3.1 Special Case
    7.3.2 General Case
    7.4 Quaternions in Matrix Form
    7.4.1 Vector Method
    7.4.2 Matrix Method
    7.4.3 Geometric Verification
    7.5 Multiple Rotations
    7.6 Eigenvalue and Eigenvector
    7.7 Rotating About an Off-Set Axis
    7.8 Frames of Reference
    7.9 Interpolating Quaternions
    7.10 Converting a Rotation Matrix to a Quaternion
    7.11 Euler Angles to Quaternion
    7.12 Summary
    7.12.1 Summary of Operations
    7.13 Worked Examples
    Chapter 8: Conclusion
    Appendix : Eigenvectors and Eigenvalues
    References
    Index
    Quaternions for Computer Graphics
    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 DOI 10.1007/978-0-85729-760-0 Springer London Dordrecht Heidelberg New York e-ISBN 978-0-85729-760-0 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 per- mitted 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 publish- ers, 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 volt- ages 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 Man- dlebrot’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. How- ever 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 becom- ing 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 were an intellectual challenge. I am now comfortable with the idea that imaginary i is nothing more than a symbol, and in the context of algebra permits i2 = −1 to be defined. And I believe it is futile trying to discover any deeper meaning to its existence. Nevertheless, it is an amazing object within mathematics, and I often wonder whether there could be similar objects waiting to be invented. When I started writing books on mathematics for computer graphics, I studied complex analysis in order to write with some confidence about complex quantities. It was then that I discovered the historical events behind the invention of vectors and quaternions, mainly through Michael Crowe’s excellent book “A History of Vector Analysis”. This book brought home to me the importance of understanding how and why mathematical invention takes place. Recently, I came across Simon Altmann’s book “Rotations, Quaternions, and Double Groups” which provided further information concerning the demise of vii
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