[.ca] Quaternions & Rotation Sequences (ISBN 0691058725)

I am the Quaternion Book's Author:
I merely want to share with you an excellent review of my Quaternion Book. The review appeared in the Nov/Dec'03 issue of Contemporary Physics, vol6., and was written by Dr Peter Rowlands, Waterloo University, UK. The review is herewith attached (if I may) otherwise I'll paste the text). It's probably too long --- but you now know where to find it. Here goes: The following Book Review Appeared in Journal: Contemporary Physics}, Nov/Dec 2003, vol 44, no. 6, pages 536 - 537 · · · Quaternions & Rotation Sequences A Primer with Applications to Orbits, Aerospace, and Virtual Reality by JACK B. KUIPERS Princeton University Press. 2002, £24.95(pbk), pp. xxii + 371, ISBN 0 691 10298 8. Scope: Text. Level: Postgraduate and Specialist. } Quaternions are one of the simplest and most powerful tools ever offered to the physicist or engineer. Unfortunately, they are relatively little known because a centuryold prejudice (the result of a family feud involving vector theory) has been responsible for keeping them out of university courses. The fact that quaternions have never really found their true role has become a self-fulfilling prophecy, despite their reappearance in various disguised forms such as Pauli matrices, 4-vectors, and, in a complex double form, in the Dirac gamma algebra. The straightforward manipulation of this relatively simple formalism, however, means that, to a quaternionist, such things as Minkowski space-time and fermionic spin are no longer mysterious unexplained physical concepts but merely inevitable consequences of the fundamental algebraic structure, while even ordinary vector algebra as David Hestenes has shown (Space-Time Algebras, Gordon and Breach, 1966) is much better understood in terms of its quaternionic base. The immense value of the quaternion algebra is that its products are ordinary algebraic products, not the dot or cross products of standard vector algebra, although they also include these concepts. Despite many statements to the contrary, quaternions are by no means short of serious applications, either. Often in highly practical contexts, and, in every application that I know of, where a quaternion formulation is possible, this formulation is invariably superior to any more 'conventional' alternative. Kuipers, in his splendid book, effectively shows this in the eminently practical case of the aerospace sequence and great circle navigation by demonstrating how the same calculations are done, first by conventional matrix methods, and then by quaternions. Rather than abstractly defining quaternion algebra and then seeking possible applications, he prepares the ground well by describing the application first, and then developing the quaternion methods which will solve it. It is not until chapter 5, in fact, that quaternion algebra is seriously introduced. However, Kuipers sets this on a firm basis by establishing early on the connection with complex numbers, matrices and rotations. These subjects are discussed with great thoroughness in the early chapters. The work is avowedly a primer, and so nothing is taken for granted. The student can begin at the beginning and follow the argument through stage by stage, with virtually no prior knowledge of the subject. The real core of the mathematical analysis comes in chapters 5 to 7, with solid and relatively easy to follow treatments of quaternion algebra and quaternion geometry, together with an algorithm summary, relating quaternions to such things as direction cosines, Euler angles and rotation operators. The superiority of quaternion over, for example, matrix methods is demonstrated by Kuipers' statement on p. 153 that the quaternion rotation operator (unlike the matrix one) is 'singularity-free'. Following the main application to the aerospace sequence and great circle navigation, there are further chapters on spherical trigonometry, quaternion calculus for kinematics and dynamics, and rotations in phase space, with two final chapters devoted to applications in electrical engineering (dipole radiation signals sent by a source to a sensor, and then correlated using a processor) and computer graphics. The final application is especially interesting as quaternions have been behind much of the rapid development of computer graphics. One role that quaternions have always fulfilled is their applicability to 3-dimensional structures, and the otherwise difficult problem of rotation, especially when time-sequencing is involved. Computer software engineers have exploited this while physicists have missed out. The creation of a 'natural' 3-dimensionality, using the 'vector' or imaginary part of quaternions was, of course, the original reason for their creation; but, while the remaining 'scalar' or real part was originally thought of as a problem by the proponents of vector theory, it is now seen as a bonus, allowing the incorporation of time as a natural result of the algebra. We cannot escape the fact that we live in time within a 3-dimensional spatial world, and quaternion algebra appears to be the easiest way of comprehending and manipulating this 3-or 4-dimension- ality. Kuipers shows us examples of the exploitation of the technique in aerodynamics, electrical engineering and computer software design, but it also has relevance in topology, quantum mechanics, and particle physics. It is frankly as absurd for physicists and engineers to neglect quaternions as it would be for them to disregard complex numbers or the minus sign. It is important that students get to learn about this spectacularly simple and powerful technique as early as possible, and Kuipers has provided us with the perfect opportunity of remedying a massive defect in our technical education. His book has everything that one could wish for in a primer. It is also beautifully set out with an attractive layout, clear diagrams, and wide margins with explanatory notes where appropriate. It must be strongly recommended to all students of physics, engineering or computer science. DR PETER ROWLANDS (University of Liverpool)

Clear and very readable:
As a "primer", this book is right on target. The theory presentation is truly legible, with remarkable notes on each page, just collateral to the main text, fixing notions or fulfilling the contents with concise demonstrations. The first chapters are a fast, but complete review of algebraic basics, good for an expert and a solid start for a novice (albeit it is instrumental to the succesive pages, refer to other specific books for consolidating your preparation). Then the volume start to unleash the true topics, showing the mathematical properties of quaternions, as a rotation operator and related geometry, to end up in a well suited exposition of spherical geometry, calculus and pertubation theory (linked to dynamics and kinematics fields). Quite pleasant to discover is the applications section, especially computer graphics, while its airspace use is just introduced. Nevertheless, the author has neatly achieved his goal.

A word of caution:
I was very disappointed when I started reading the book and immediately noticed a number of errors in the formulae. These were most likely typos but still can be confusing at times. As the other reviewers mentioned, the book has a very interesting, and in my opinion very good, teaching sytle, but don't take all the formulae by heart if you are going to use it as a reference book. (PS. My comments are on the first print of the book, I hope the errors have been corrected in the later prints.)

A good introduction to quaternions:
Is it possible to recommend a book and still say that it needs revision? It needs revision precisely because it is a good book and may well find more readers. The book does what no other does as far as I know; it introduces quaternions in elementary terms and shows some, at least, of how useful the concept is. The topic is neglected in textbooks for students at this level and probably even more generally. And yet I do think that the author could revise this book substantially and produce a better one.

A Delightful Read!:
This book was a delightful read! If you ever have been curious or puzzled or even terrified by Euler angles then read this text. Many questions will be answered and much knowledge revealed. For a gentle introduction to quaternions this is also a good place to start. The book starts out with a review of complex numbers (in order to emphazise the similarity to quaternions later on), then reviews rotations and matrix methods (sorry but vectors don't do rotations) and then gets into the nitty-gritty of rotations in 2-space and on into 3-space. Three problems involving rotations are discussed in detail. All of this at first with matrix methods and then a nice easy introduction to quaternions is given and these three problems are then handled with quaternions. There is a strong comparison made between compex number arithmetic and quaternion arithmetic, such as norms, conjugates and computation of multiplicative inverses. Ever wonder how far it is between say Dallas and London? And what direction to take to go from to the other? Well, airplanes do it every day but if I were asked that question on an exam I would have flunked it. Not anymore! The explanation of the answer to such questions is presented in a simple/y delightful manner in this text. There is also stuff here on spherical trigonometry and a description of an orientation and distance sensing system, using the near field pattern of magnetic dipole antennas. Finally there is discussion of ordinary differential equations and an overview of what is needed for displaying moving objects with computer graphics. Well, that is quite a lot, but the pace is easy going and the text takes this into account by reproducing say the equation or the figure under discussion in the margins as it goes along. A very well executed text, no constant back-paging to figure out what we were talking about! The text has the flavor being written from lecture notes, not the usual cryptic ones, but well expanded and well thought out ones. This leads to some repetition but that's O.K. by me. It makes easy reading for a varied audience. Who is this text aimed at? Well I did find it enlightening even with a background in physics and a rudimentary introduction to Euler angles in an advanced classical mechanics course, but I never had the occasion to use them in my career, so this was a good refresher for me. What do you need to know to get something out of this text? A good grip on the meaning of sines and cosines and the various addition and multipication formulas or at least know where to look them up. A little knowledge of vectors, the dot and cross product will also be handy even though it is explained in the text. For one chapter a smattering of differential calculus is useful and for another a whole lot of knowledge about differential equations, more than I have is needed. But if you don't have this background you can safely skip these parts and not loose any of the further stuff in the text. You should know how to solve sets of simultaneous equations, inhomogeneous and homogeneous. Matrix operations are all discussed in detail and you can learn them here. You will probably get one of the best introductions to the concept of eigenvectors that you can find anywhere, something that will stick with you for the rest of your career. Well who is it aimed at? Anyone interested in spherical metrology, astronomy, robotics, orbital mechanics, graphical stuff, classical mechanics and so on. A smart high school student could learn a lot here and anyone with a few years of college math/science under his belt will find it profitable as will some, like me, with an advanced degree but no detailed experience in this field. What did I miss in this text? You know how you visualize two component complex numbers as points in the plane and you might think that a 3 component entity might do the same thing with points in 3 dimensional space. Not so if you want it to be an algebra says Frobenius, as mentioned in the book. But there is a short (half page) demonstration that a 3 component hyper-complex number with real coefficients leads immediately to a logical contradiction (e.g. Simmons, Calculus Gems.) This demo would reinforce the need for 4 component quaternions. Why do quaternions describe a rotation in terms of the half angle? Well maybe because you need a quaternion and its conjugate both to describe the rotation. But to me there is an even better source for this oddity, namely the description of a rotation as two successive relections. Then the origin of half angles shines right out of the geometry (e.g. Snygg, Cilfford Algebra, a 2-3 page description in Chapter 1. Also find here a solution to the spinning top problem using quaternion calculus.) Quaternions do simplify the derivation of many formulas but do they speed up the numerical computations? There is no real discussion of this topic. It might take a couple of chapters and you need to quit somewhere I guess. Criticsisms?. No, merely matters of taste. The final chapter treats the more general motion of a body: rotations, translations, scaling, perspective and sensivity factors. Here we run into the puzzle that all this can be easily handled with matrix methods but apparently not with quaternions. The question then arises why bother with quaternions at all, at least for numerical work. I found the text here a little weak. A criticism that I do have is the definition by the author of the reversal of the vector part of the quaternion as its complex conjugate. One property of this conjugate is that the conjugate of the product of two quaternions is the product of the conjugates in reverse order. But this is not true of the usual complex conjugate, the compex conjugate of the product of two matrices, say, is the product of the complex conjugates of each matrix but in the same order. Does this lead to problems in this text? No, complex numbers and matrices or quaternions are never mixed here. But the idea can lead a novice astray in future work. At any rate this is a great text with no typos in the many formulas that I could detect. As I said a Great Read.