Quaternions for Computer Graphics

【实例简介】
这是我找到的关于解释Quaternion（四元数）最好的资料了。从复数表示旋转开始，一步步的深入到如何使用四元数表示旋转，如何绕任意轴旋转。深入浅出，明了易懂。（暂时没有中文版，有时间的时候试试翻译下）
John vince Quaternions for computer Graphics S ringer Professor john vince MTech. PhD. DSc CEng fbcs Bournemouth University, Bournemouth, UK urlwww.johnvince.co.uk ISBN978-0-85729-7594 e-ISBN978-0-85729-760-0 DOI10.1007/9780-85729-760-0 Springer london dordrecht heidelberg new york ritish Library Cataloguing in Publication Data A catalogue record for this book is available from the British library Library of Congress Control Number: 2011931282 o 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 us 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 SpringerispartofSpringerScience+businessMedia(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 engineeringI 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 i I to be defined, and i believe t is futile trying to discover any deeper meaning to its existence. Nevertheless, it i 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 Crowes 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 Preface quaternions in the 19th century Altmann is very passionate about securing recogni tion for the mathematical work of Olinde rodrigues, who published a formula that s very similar to that generated by Hamilton's quaternions. The important aspect of Rodrigues' publication was that it was made three years before Hamiltons invention of quaternions in 1843. However, Rodrigues did not invent quaternion algebra--that prize must go to Hamilton--but he did understand the importance of half-angles in the trigonometric functions used to rotate points about an arbitrary axis Anyone who has used Euler transforms will be aware of their shortcomings, espe cially their Achilles'heel: gimbal lock. Therefore, any device that can rotate points about an arbitrary axis is a welcome addition to a programmer's toolkit. There are many techniques for rotating points and frames in the plane and space, which I covered in some detail in my book Rotation Transforms for Computer graphics That book also covered the Euler-Rodrigues parameterisation and quaternions, but it was only after submitting the manuscript for publication, that I decided to write this book dedicated to quaternions and how and why they were invented, and their application to computer graphics Whilst researching this book, it was extremely instructive to read some of the early books and papers by william Rowan Hamilton and his friend P.G. Tait. I now understand how difficult it must have been to fully comprehend the significance of quaternions, and how they could be harnessed. At the time, there was no major demand to rotate points about an arbitrary axis however, a mathematical system was required to handle vectorial quantities. In the end, quaternions were not the flavour of the month and slowly faded from the scene Nevertheless the ability to represent vectors and manipulate them arithmetically was a major achievement for hamilton even though it was the foresight of Josiah Gibbs to create a simple and workable algebraic framework In this book i have tried to describe some of the history surrounding the invention of quaternions, as well as a description of quaternion algebra. In no way would I consider myself an authority on quaternions. I simply want to communicate how I understand them, which hopefully will be useful for you. There are different ways to represent a quaternion, but the one I like the best is an ordered pair, which I discovered in simon altmann's book This book divides into eight chapters. The first and last chapters introduce and conclude the book, with six chapters covering the following subjects. The second chapter on number sets and algebra reviews the notation and language relevant to the rest of the book. There are sections on number sets, axioms, ordered pairs, groups rings and fields. This prepares the reader for the non-commutative quaternion prod uct,and why quaternions are described as a division ring Chapter 3 reviews complex numbers and shows how they can be represented as an ordered pair and a matrix. Chapter 4 continues this theme by introducing the complex plane and showing the rotational features of complex numbers. It also prepares the reader for the question that was asked in the early nineteenth century could there be a 3D equivalent of a complex number? Chapter 5 answers this question by describing Hamilton's invention: quaternions and their associated algebra i have included some historical information so that the Preface reader appreciates the significance of Hamiltons work. Although ordered pairs are the main form of notation i have also included matrix notation To prepare the reader for the rotational qualities of quaternions, Chap 6 reviews 3D rotation transforms, especially Euler angles, and gimbal lock. I also develop a matrix for rotating a point about an arbitrary axis using vectors and matrix trans- forms Chapter 7 is the focal point of the book and describes how quaternions rotate vec tors about an arbitrary axis The chapter begins with some historical information and explains how different quaternion products rotate points. Although quaternions are readily implemented using their complex form or ordered-pair notation, they also have a matrix form, which is developed from first principles. The chapter continues with sections on eigenvalues, eigenvectors, rotating about an offset-axis, rotating frames of reference, interpolating quaternions, and converting between quaternions and a rotation matrix Each chapter contains many practical examples to show how equations are eval- uated and where relevant, further worked examples are shown at the end of the Writing this book has been a very enjoyable experience, and i trust that you will also enjoy reading it and discover something new from its pages I would like to thank Dr Tony Crilly, Reader Emeritus at Middlesex University, for reading a draft manuscript and correcting and clarifying my notation and expla nations. Tony performed the same task on my book Rotation Transforms for Com puter Graphics. I trust implicitly his knowledge of mathematics and I am grateful for his advice and expertise. However, I still take full responsibility for any algebraic faux pas I might have made I would also like to thank professor Patrick riley, who read some early drafts of the manuscript and posed some interesting technical questions about quaternions Such questions made me realise that some of my descriptions of quaternions re- quired further clarification, which hopefully have been rectified ti i have now used IATEX28 for three of my books. and have become confident with notation. Nevertheless, I still had to call upon Springers technical support team, and thank them for their help I am not sure whether this is my last book. If it is, I would like to thank beverley Ford, Editorial Director for Computer science and helen desmond, Associate ed- itor for Computer Science, Springer UK, for their professional support during the past years. If it is not my last book, then I look forward to working with them again on another project Ringwood. UK JohnⅤince Contents 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 2 Number Sets and Algebra 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 44456778800 2.8.2 Abelian Group 2.8.3 Rings 2.8.4 Fields 10 2.8.5 Division ring 2.9 Summar 2.9.1 Summary of definitions 11 3 Complex Numbers 13 3.1 Introduction 3.2 Imaginary Numbers 3.3 Powers of i Contents 3.4 Complex Numbers .15 3.5 Adding and subtracting complex numbers 16 3.6 Multiplying a Complex Number by a Scalar 16 3.7 Complex Number Products 16 3.7.1 Square of a Complex Number 17 3.8 Norm of a Complex number 3.9 Complex Conjugate 18 3. 10 Quotient of Two Complex Numbers 18 3.11 Inverse of a Complex Number 3. 12 Square-Root of 3.13 Field Structure 21 3.14 Ordered Pairs 21 3.14.1 Multiplying by a scalar 22 3. 14.2 Complex Conjugate 3. 14.3 Quotient 3. 14.4 Inverse 23 3.15 Matrix Representation of a complex number 24 3.15.1 Adding and subtracting 3.15.2 The product 25 3. 15.3 The Square of the Norm 3.15.4 The Complex conjugate 25 3. 15.5 The Inverse 26 3. 15.6 Quotient .26 3.16 Summary 27 3.16.1 Summary of operations 3. 17 Worked Examples 29 4 The Complex plane 33 4.1 Introduction 4.2 Some history 4.3 The Complex plane .34 4.4 Polar Representation 37 4.5 Rotors 4.6 Summary 4.6. 1 Summary of operations .42 4.7 Worked Examples 5 Quaternion algebra 47 5.1 Introduction 47 5.2 Some history 49 5.3 Defining a Quaternion 53 5.3. 1 The Quaternion Units 5.3.2 Example of Quaternion Products .56 5.4 Algebraic Definition 56 5.5 Adding and Subtracting Quaternions 57 5.6 Real Quaternion ...57 【实例截图】
【核心代码】