Quaternions & Rotation Sequences [Hardcover]

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)

This book is simply EXCELLENT.
Its goal, namely, to convey quaternion algebra to people not necessarily mathematicians, is really fulfilled.
A variety of readers should benefit from this work.

And, hopefully, quaternions will soon become part of conventional mathematics education,
as well as part of every branch of Science - including, for instance, biology and medicine.

Dr. Kuipers' "Quaternions and Rotation Sequences" is a fundamental step in this direction.
It presents, elegantly and authoritatively, this unequaled, powerful algebraic system, initially proposed by Sir William R. Hamilton in 1843.
It is surprising just how long Hamilton's Quaternions have been forgotten...

This book is hence a BREAKTHROUGH, not only in mathematics, but in all Science.

If you've never heard about "quaternions" before, yet science and its achievements are of interest to you,
don't hesitate to dig more on this topic, and this book is sure one of the first steps!
And if you did hear about quaternions, maybe you have heard that vectorial algebra was equivalent and more tractable...
That's NOT TRUE. Quaternions are far more superior than vector algebra, and this book shows us why.

I first heard about quaternions just a couple of years ago, through Wikipedia, by accident, while searching "gyroscopes"...
It struck me a lot to read that there are numbers with 4 dimensions; with 4 inherent dimensions!!!
Immediately, I knew this had to be an UTTERLY IMPORTANT scientific field.
Yet, what surprised me most was that I had never heard about it, ever. How could this be?
Not any of my math teachers,
neither any of the mathematics nor philosophy of science books that I've read, or that I happened to look a while,
none have never even mentioned its existence!!!

That puzzled me for a while, and I thought that maybe quaternion algebra might be extremely difficult for mathematicians and non-mathematicians alike.
Fortunately, Dr. Kuipers' book proved me I was wrong: Quaternions are just as accessible as any other conventional mathematics.
Eventually, it is even easier, were it presented to us early, as part as our ordinary mathematical education in schools.

So, now, this is one of MY MOST CHERISHED BOOKS, and it figures as one of those few turning-point books that I have read,
delimitating a "before-and-after" in my personal history.
(By the way, my other turning-point book,s are:
"À la Recherche du Temps Perdu [In Search of Lost Time]", by Proust,
"The Structure of Scientific Revolutions", by Thomas Khun;
"Darwinism Evolving: Systems Dynamics and the Genealogy of Natural Selection", by David Depew and Bruce Weber;
"L'Évolution Créatrice" (Creative Evolution), by Henry Bergson;
"Critique of Pure Reason", by Kant; and
"Mind in Life: Biology, Phenomenology, and the Sciences of Mind" by Evan Thompson).

Unquestionably, it is scientific masterpiece.

alo_world

I bought this book a few years ago and it was one of the best purchases I ever made. I spent a few months trying to write my own matrix based code for rotations. Then I spent a few months trying to implement quaternion rotation with incorrect formulas I kept finding online. I finally got desperate and bought this book and everything improved over night..I had rotating objects, rotating cameras and ways to calculate deflections and accumulating rotations. This is a math book, not a programming book. But if you need to implement quaternions in a program, the first part of the book will give you formulas you need (no calculus) with a baby spoon.