QuaternionsforComputerGraphics

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 vinceQuaternionsfor computerGraphicsSringerProfessor john vince MTech. PhD. DScCEng fbcsBournemouth University, Bournemouth, UKurlwww.johnvince.co.ukISBN978-0-85729-7594e-ISBN978-0-85729-760-0DOI10.1007/9780-85729-760-0Springer london dordrecht heidelberg new yorkritish Library Cataloguing in Publication DataA catalogue record for this book is available from the British libraryLibrary of Congress Control Number: 2011931282o Springer-Verlag London Limited 2011Apart 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 reproducedstored 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 theCopyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent tothe publishersThe use of registered names, trademarks, etc, in this publication does not imply, even in the absence of aspecific statement, that such names are exempt from the relevant laws and regulations and therefore freefor general usThe publisher makes no representation, express or implied, with regard to the accuracy of the informationcontained in this book and cannot accept any legal responsibility or liability for any errors or omissionsthat may be madeCover design: VTeX UAB, lithuaniaPrinted on acid-free paperSpringerispartofSpringerScience+businessMedia(www.springer.com)This book is dedicated to heidiPrefaceMore than 50 years ago when I was studying to become an electrical engineerI 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, ratherthan i, because the latter stood for electrical current. So from the very start of mystudies I had a clear mental picture of the imaginary unit as a rotational operatorwhich could advance or retard electrical quantities in timeWhen events dictated that I would pursue a career in computer programming-rather than electrical engineeringI had no need for complex numbers, until Mandlebrot's work on fractals emerged. But that was a temporary phase, and I neverneeded 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 internalreport on quaternions, which were being used to control the rotational orientation ofa simulated aircraftI can still remember being completely bemused by quaternions simply becausethey involved so many imaginary terms. However, after much research i started tounderstand 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 termswith the real meaning? of mathematics through the writing of Bertrand russellConsequently, concepts such as i were an intellectual challengeI am now comfortable with the idea that imaginary i is nothing more than asymbol, and in the context of algebra permits iI to be defined, and i believet is futile trying to discover any deeper meaning to its existence. Nevertheless, it ian amazing object within mathematics and i often wonder whether there could besimilar objects waiting to be inventedWhen I started writing books on mathematics for computer graphics, I studiedcomplex analysis in order to write with some confidence about complex quantitiesIt was then that i discovered the historical events behind the invention of vectors andquaternions, mainly through Michael Crowes excellent book"A History of vectorAnalysis". This book brought home to me the importance of understanding how andwhy mathematical invention takes placeRecently, I came across Simon Altmann's book " Rotations, Quaternions, andDouble groups"which provided further information concerning the demise ofPrefacequaternions in the 19th century Altmann is very passionate about securing recognition for the mathematical work of Olinde rodrigues, who published a formula thats very similar to that generated by Hamilton's quaternions. The important aspect ofRodrigues' publication was that it was made three years before Hamiltons inventionof quaternions in 1843. However, Rodrigues did not invent quaternion algebra--thatprize must go to Hamilton--but he did understand the importance of half-angles inthe trigonometric functions used to rotate points about an arbitrary axisAnyone who has used Euler transforms will be aware of their shortcomings, especially their Achilles'heel: gimbal lock. Therefore, any device that can rotate pointsabout an arbitrary axis is a welcome addition to a programmer's toolkit. There aremany techniques for rotating points and frames in the plane and space, which Icovered in some detail in my book Rotation Transforms for Computer graphicsThat book also covered the Euler-Rodrigues parameterisation and quaternions, butit was only after submitting the manuscript for publication, that I decided to writethis book dedicated to quaternions and how and why they were invented, and theirapplication to computer graphicsWhilst researching this book, it was extremely instructive to read some of theearly books and papers by william Rowan Hamilton and his friend P.G. Tait. I nowunderstand how difficult it must have been to fully comprehend the significanceof quaternions, and how they could be harnessed. At the time, there was no majordemand to rotate points about an arbitrary axis however, a mathematical system wasrequired to handle vectorial quantities. In the end, quaternions were not the flavourof the month and slowly faded from the scene Nevertheless the ability to representvectors and manipulate them arithmetically was a major achievement for hamiltoneven though it was the foresight of Josiah Gibbs to create a simple and workablealgebraic frameworkIn this book i have tried to describe some of the history surrounding the inventionof quaternions, as well as a description of quaternion algebra. In no way would Iconsider myself an authority on quaternions. I simply want to communicate how Iunderstand them, which hopefully will be useful for you. There are different waysto represent a quaternion, but the one I like the best is an ordered pair, which Idiscovered in simon altmann's bookThis book divides into eight chapters. The first and last chapters introduce andconclude the book, with six chapters covering the following subjects. The secondchapter on number sets and algebra reviews the notation and language relevant to therest of the book. There are sections on number sets, axioms, ordered pairs, groupsrings and fields. This prepares the reader for the non-commutative quaternion product,and why quaternions are described as a division ringChapter 3 reviews complex numbers and shows how they can be representedas an ordered pair and a matrix. Chapter 4 continues this theme by introducingthe complex plane and showing the rotational features of complex numbers. It alsoprepares the reader for the question that was asked in the early nineteenth centurycould there be a 3D equivalent of a complex number?Chapter 5 answers this question by describing Hamilton's invention: quaternionsand their associated algebra i have included some historical information so that thePrefacereader appreciates the significance of Hamiltons work. Although ordered pairs arethe main form of notation i have also included matrix notationTo prepare the reader for the rotational qualities of quaternions, Chap 6 reviews3D rotation transforms, especially Euler angles, and gimbal lock. I also develop amatrix for rotating a point about an arbitrary axis using vectors and matrix trans-formsChapter 7 is the focal point of the book and describes how quaternions rotate vectors about an arbitrary axis The chapter begins with some historical information andexplains how different quaternion products rotate points. Although quaternions arereadily implemented using their complex form or ordered-pair notation, they alsohave a matrix form, which is developed from first principles. The chapter continueswith sections on eigenvalues, eigenvectors, rotating about an offset-axis, rotatingframes of reference, interpolating quaternions, and converting between quaternionsand a rotation matrixEach chapter contains many practical examples to show how equations are eval-uated and where relevant, further worked examples are shown at the end of theWriting this book has been a very enjoyable experience, and i trust that you willalso enjoy reading it and discover something new from its pagesI would like to thank Dr Tony Crilly, Reader Emeritus at Middlesex University,for reading a draft manuscript and correcting and clarifying my notation and explanations. Tony performed the same task on my book Rotation Transforms for Computer Graphics. I trust implicitly his knowledge of mathematics and I am gratefulfor his advice and expertise. However, I still take full responsibility for any algebraicfaux pas I might have madeI would also like to thank professor Patrick riley, who read some early drafts ofthe manuscript and posed some interesting technical questions about quaternionsSuch questions made me realise that some of my descriptions of quaternions re-quired further clarification, which hopefully have been rectifiedti i have now used IATEX28 for three of my books. and have become confident withnotation. Nevertheless, I still had to call upon Springers technical support team,and thank them for their helpI am not sure whether this is my last book. If it is, I would like to thank beverleyFord, Editorial Director for Computer science and helen desmond, Associate ed-itor for Computer Science, Springer UK, for their professional support during thepast years. If it is not my last book, then I look forward to working with them againon another projectRingwood. UKJohnⅤinceContents1 Introduction1.1 Rotation transforms1. 2 The reader1.3 Aims and objectives of This book1. 4 Mathematical Techniques1.5 Assumptions made in This book2 Number Sets and Algebra2.2 Number sets2.2.1 Natural numbers2.2.2 Real Numbers2.2.3 Integers2.2 4 Rational numbers2.3 Arithmetic Operations2.4 Axioms2.5 Expressions2.6 Equations2.7 Ordered Pairs2. 8 Groups, Rings and Fields2.8.1 Groups444567788002.8.2 Abelian Group2.8.3 Rings2.8.4 Fields102.8.5 Division ring2.9 Summar2.9.1 Summary of definitions113 Complex Numbers133.1 Introduction3.2 Imaginary Numbers3.3 Powers of iContents3.4 Complex Numbers.153.5 Adding and subtracting complex numbers163.6 Multiplying a Complex Number by a Scalar163.7 Complex Number Products163.7.1 Square of a Complex Number173.8 Norm of a Complex number3.9 Complex Conjugate183. 10 Quotient of Two Complex Numbers183.11 Inverse of a Complex Number3. 12 Square-Root of3.13 Field Structure213.14 Ordered Pairs213.14.1 Multiplying by a scalar223. 14.2 Complex Conjugate3. 14.3 Quotient3. 14.4 Inverse233.15 Matrix Representation of a complex number243.15.1 Adding and subtracting3.15.2 The product253. 15.3 The Square of the Norm3.15.4 The Complex conjugate253. 15.5 The Inverse263. 15.6 Quotient.263.16 Summary273.16.1 Summary of operations3. 17 Worked Examples294 The Complex plane334.1 Introduction4.2 Some history4.3 The Complex plane.344.4 Polar Representation374.5 Rotors4.6 Summary4.6. 1 Summary of operations.424.7 Worked Examples5 Quaternion algebra475.1 Introduction475.2 Some history495.3 Defining a Quaternion535.3. 1 The Quaternion Units5.3.2 Example of Quaternion Products.565.4 Algebraic Definition565.5 Adding and Subtracting Quaternions575.6 Real Quaternion...57