Customer Reviews

Quaternions and Rotation Sequences

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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)

Latitude and longitude look simple enough, at first - just put your finger in the globe, and see which horizontal line crosses which vertical. When you start doing arithmetic, though, things get weird. Measuring longitude in degrees, 179+2=-179. In degrees latitude, 89+2=89, but the longitude changes! And, when you try to figure longitude precisely at the north pole, you run into a singularity. Believe me, you don't want to be in a plane when its navigation programs run into singularities.

Those bits of strangeness all vanish when quaternions represent angles. Quaternions are a bit like complex numbers, but with three different complex parts instead of one. They have very nice mathematical properties, even better than rotation matrices, and a compact form.

Kuipers gives a clear, thorough introduction to quaternions and their uses in geometric computations. Everything is explained one step at a time, giving the reader plenty of chance to back off and try again when the discussion gets thick. The buildup is very methodical, just about every derivation is carried out in steps that are easy to follow, using legible, standard notation. Kuipers uses side bars to remind the reader about the basics under more complex discussions, keeping an awareness of where a beginner might go off the rails. Since this discusses geometric computations, illustrations are profuse.

The book is not for the reader in a hurry. There are lots of gems here, but you really do have to dig through a lot to find them. The illustrations contain all needed information, but it may take some effort to pick them apart. And, like any technical book, this assumes a reader with a certain background. In this case, intuition about 3D objects, trig, and linear algebra are compulsory, but I guess a sufficiently dedicated reader could substitute blind obedience to formulas for linear algebra. Ch.11-13 assumes calculus through partial differentials and ODEs, but many readers can skip these chapters without loss.

This is all the how and why of quaterion representations of 3D rotations. It's gently paced, and makes only moderate assumptions about the reader's background. I've never seen this material presently so clearly, from so many angles, anywhere else. Highly recommended.

//wiredweird

My graduate school work was in theoretical quantum mechanics, and was especially concentrated in the group properties of rotations. I can honestly say that I would have been twice as effective if I had this reference available then.

Kuiper does an outstanding job of pulling together the traditional matrix-based approach to describing rotations with the less-frequently encountered quaternion approach. In doing so, he clearly shows the benefits of the quaternion algebra, especially for computer systems modeling rigid body rotations and virtual worlds. The exposition is clear, concise, and aimed at the practitioner rather than the theoretician. The examples are taken from classical engineering problems -- a refreshing change from the quantum-mechanical problems I was used to from previous works on the subject.

Despite the practical foocus, though, there is plenty of material here for those more interested in understanding the minutia of the SO(3) symmetry group. And unlike most work in this field, he doesn't stop with algebra, but includes the calculus of rotation matrices and quaternions using material on kinematics and dynamics of rigid bodies, celestial mechanics, and rotating reference frames.

I give the book my highest recommendation. It should be considered an essential reference work for anyone who encounters rotational problems with any frequency.

--Tony Valle

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.

I am mostly self educated in mathematics but still had no trouble with the reasoning and topics in this book. Each topic is intuitively and rigorously explained. Quaternions are a delight, are very interesting to work with, and are suprisingly productive in use. It is hard to find a good solid text on quaternions so this book would be well appreciated to anyone interested in the subject. Brush up on your matrix algebra first, especially determinants. I would recommend this book to anyone interested in applied mathematics.

This book is one of the very few mathematical, engineering, physics texts that I have come across in my 37 year career that stands out as a classic to be. I read the entire book and because it is so good, it is one of the few that I actually retained.
The book SAVES TIME. It saves time because you can learn the material without second guessing what the author is trying to say, because he wrote the book to TEACH you the material, not so you could stand in awe at his skill.
Because he took this topic and made it understandable in a way that so few books of its type do, his skill is evident in his clarity.
Anyone who can teach a subject like this in a book, with no classroom instruction knows his subject cold.
This is a winner. Big time.

I want to personally thank the author for taking the time and pains to write his book for me and countless others who stumble around in the world of physics, mathematics and engineering trying to get a grasp on our massive field. His contribution shines.

This book is one of the most understandable and down-to-earth mathematics texts I've ever read. For instance, after presenting a new concept, he'll summarize it again in the sideline of the book every time he refers to it for the next twenty pages or so. At first, I was finding myself getting annoyed, and thinking, "What, does he think I'm stupid?"

Then I considered the alternative, the terse style of so many mathematical texts that has me regularly flipping between eight different pages trying to put everything together. I stopped complaining and started appreciating Kuipers' approach.

Kuipers does assume a certain amount of familiarity with mathematics, but not any knowledge in particular, as he reviews basic matrix multiplication and the like at the beginning of the book.

For a topic that can seem daunting (our artist always makes fun of me using seemingly gratuitous big phrases like "spherically interpolated quaternion splines") this book makes it very understandable. If you need to work with computational rotation, for a flight sim, robotics visualization, or (most importantly) for a computer game, I can't recommend this book highly enough!

This book is great. The author goes into great depth on Matrices and Quaternions. Topics such as Aerospace sequences and Tracking sequences are covered in clear detail.

This book is a must have for anyone taking a Computer Graphics course. If you have ever studied Shoemake's arcball then you will appreaciate the Tracking sequence.

It is just too bad other authors don't accumulate this much information and present it in a clear usuable format.

I give this book a very high recommendation. You will not find better information than this on quaternion and/or rotation sequences.

The main asset of this delightful book is its methodical and unencumbered presentation of the most basic mathematics, vector and matrix operations from the first page. Specifically, it illustrates basic algebraic field theory and generalizes complex numbers into quaternions in an uncomplicated way. The fluid presentation encourages the reader to continue through the necessarily lengthy introduction of the classical rotation operators (as detailed use of quaternions doesn't start until about 100 pages, in Chapter 5).

I appreciated the fact its introductory nature is honestly clarified by the subtitle: it is a self-declared primer. It is also one of the few textbooks I have seen making extensive use of a marginal gloss (explanatory notes in the margin), which seems much more efficient than footnotes or appendices. Many facts are repeated - noticeable but not too annoying, and handled well in the gloss. This level of presentation will certainly benefit most readers new to the subject. Anyone writing a technically oriented textbook should consider reviewing this title for its format alone.

The book defines a quaternion as a 3-D vector plus a scalar. Defining the quaternion with these more conventional mathematical notions makes the very concept more approachable. But it is not clear whether this (and other) notation is truly unique to this book or otherwise widely acknowledged in literature. For example, most of the notation adopted for classic rotation operators seemed unnecessarily different (and therefore slightly confusing) compared to those few other engineering and science textbooks I've been able to reference on the subject. And a few terms, such as "kyperplane", appear unique to this book alone.

Considering that this is an introductory textbook, the recommended "further reading" list was by far the most disappointing aspect of this title. Out of sixteen (16) meager references provided, 1/4 are Prof. Kuipers' own patent declarations; the rest are mostly hard-to-get Air Force reports, out of print books, and a few specialty journal articles. The lack of specific references is especially bothersome when facts or theorems are cited without support or proof, such as "Euler's Theorem" (p. 83).

Engineers and engineering students should also be aware that some of the "applications to orbits and aerospace" (from the subtitle) appear to be more for academic or illustrative purposes than for immediate, practical application. For example, the publisher's on-line table of contents identifies "Chapter 11 - Quaternion Calculus for Kinematics and Dynamics." However, this chapter doesn't really cover the conventional transformations of relative velocity or accelerations with respect to rotating frames of reference, which is essential to the study of dynamics and kinematics of air and space vehicles. In the preface, the author acknowledges that "It was difficult knowing where to stop, since the subject deserves much more attention and greater depth." As a result, the book may have slightly more appeal to those interested in 3-D programming and visualization.

God bless the author, who at age 80 apparently supplied the textbook copy in camera ready form. Unfortunately, my 3rd printing still contains many obvious typographical errors, which is the publisher's responsibility (who holds the copyright). A lack of editorial review normally implies that less obvious errors are lurking in those all-important equations, but thankfully Prof. Kuipers is kind enough to provide an errata sheet if the reader requests it via email. However, the reader should be aware that his printed book is still be published uncorrected, and no official errata appears at the publisher's website at this time.

In summary, I would recommend this primer for the engineering student or programmer with a novice to intermediate level of familiarity with rotational sequences. The book's style of presentation is commendable, and the extensive gloss makes the subject matter more understandable to the beginner. Discussions of some engineering applications, as well as specific topics such as orbital mechanics, gravitational theory, etc., are presented with far less detail, clarity, and rigor. While disappointing, this is forgivable as the author seemingly intends to illustrate, rather than develop rigorously complete relationships, for these applications. However, the lack of modern, easily obtained references and some seemingly unique notation may give this title less longevity as a research or reference text.

This book is a masterpiece. It has a beautiful blending of both the theory and the practical. An abstract subject has been transformed by an expert with many new and heretofore unused explications. I highly recommend that this book be studied from cover to cover by students of Mathematical Physics and many fields of Engineering. The halls of academia should incorporate this great work into their courses. I also recommend that MathCAD and Mathematica embrace the material into their tools with the permission of the author. I would be happy to offer some suggestions for such a venture.