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Quaternions and Rotation Sequences: A Primer With Applications to Orbits, Aerospace and Virtual Reality [Anglais] [Relié]

J.B. Kuipers

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Amazon.com: 4.8 étoiles sur 5  38 commentaires
64 internautes sur 65 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 I am the Quaternion Book's Author 25 janvier 2004
Par J.B. Kuipers - Publié sur Amazon.com
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
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.

(University of Liverpool)
47 internautes sur 47 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 Plainest, clearest introduction around 19 juin 2005
Par wiredweird - Publié sur Amazon.com
Format:Broché|Achat authentifié par Amazon
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.

57 internautes sur 59 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 An oustanding work on rotations for the practitioner 1 mai 1999
Par Tony Valle - Publié sur Amazon.com
Format:Relié|Achat authentifié par Amazon
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
15 internautes sur 15 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 A Delightful Read! 8 mars 2002
Par M de Wit - Publié sur Amazon.com
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

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.
10 internautes sur 10 ont trouvé ce commentaire utile 
5.0 étoiles sur 5 smooth read. 23 septembre 2005
Par James M. Pothering - Publié sur Amazon.com
Format:Broché|Achat authentifié par Amazon
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.
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