Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace and Virtual Reality [ハードカバー]

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