Geometric Algebra for Computer Graphics [Hardcover]

John A. Vince (Author)

Book Description

ISBN-10: 1846289963 | ISBN-13: 978-1846289965 | Publication Date: April 14, 2008 | Edition: 1st Edition.

Geometric algebra (a Clifford Algebra) has been applied to different branches of physics for a long time but is now being adopted by the computer graphics community and is providing exciting new ways of solving 3D geometric problems. The author tackles this complex subject with inimitable style, and provides an accessible and very readable introduction. The book is filled with lots of clear examples and is very well illustrated. Introductory chapters look at algebraic axioms, vector algebra and geometric conventions and the book closes with a chapter on how the algebra is applied to computer graphics.

Editorial Reviews

From the Back Cover

Since its invention, geometric algebra has been applied to various branches of physics such as cosmology and electrodynamics, and is now being embraced by the computer graphics community where it is providing new ways of solving geometric problems. It took over two thousand years to discover this algebra, which uses a simple and consistent notation to describe vectors and their products. John Vince (best-selling author of a number of books including ‘Geometry for Computer Graphics’ and ‘Vector Analysis for Computer Graphics’) tackles this new subject in his usual inimitable style, and provides an accessible and very readable introduction. The first five chapters review the algebras of real numbers, complex numbers, vectors, and quaternions and their associated axioms, together with the geometric conventions employed in analytical geometry. As well as putting geometric algebra into its historical context, John Vince provides chapters on Grassmann’s outer product and Clifford’s geometric product, followed by the application of geometric algebra to reflections, rotations, lines, planes and their intersection. The conformal model is also covered, where a 5D Minkowski space provides an unusual platform for unifying the transforms associated with 3D Euclidean space. Filled with lots of clear examples and useful illustrations, this compact book provides an excellent introduction to geometric algebra for computer graphics.

Product Details

• Hardcover: 272 pages
• Publisher: Springer; 1st Edition. edition (April 14, 2008)
• Language: English
• ISBN-10: 1846289963
• ISBN-13: 978-1846289965
• Product Dimensions: 9.4 x 7 x 0.7 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 3.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #2,120,698 in Books (See Top 100 in Books)

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14 of 14 people found the following review helpful:

3.0 out of 5 stars A nice introduction, but be careful!, June 15, 2010

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J. J. K. Swart (Den Haag, ZH Netherlands) - See all my reviews
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This review is from: Geometric Algebra for Computer Graphics (Hardcover)

The title does no justice to this book. It is a nice introduction to Geometric Algebra for anyone, not just computer graphics programmers. It shows clearly, that Geometric Algebra is a better alternative to Vector Algebra, which is the mainstream method of dealing with 2 and 3 dimensional geometry.

Geometric Algebra is a far more logical way to deal with geometry than vector algebra. Vector algebra tries to represent geometry by building up everything from points. In Geometric Algebra you have more tools. In fact, you have exactly as many tools as there are kinds of basic geometric objects. In Geometric Algebra you have mathematical representations of points, lines, planes and volumes, exactly the things that come to mind if you think about geometry.

This allows, for example, for a far more logical description of kinematics. To give an example, within 3 dimensional vector algebra a rotation is represented by a vector perpendicular to the plane of rotation, formed out of a cross product.

Within Geometric Algebra a rotation is represented by ... the rotation itself! How? through the introduction of a bivector that defines the plane of rotation. The size of this bivector then represents the magnitude of the rotation. This is much more logical than artificially constructing a vector perpendicular to the rotation. In fact, the vector method often fails, not only within 4 dimensions, but already in 2 dimensions. You cannot represent a rotation in 2 dimensions with a vector perpendicular to the plane of rotation, because, in 2 dimensions there are not enough dimensions to do that. Moreover, it creates a wrong habit. In 3 dimensions you have 3 translation directions and 3 rotational directions. This creates the false illusion that, in general, the number of rotational degrees of freedom in any dimension is equal to the number of translational degrees of freedom. This is already false in 2 dimensions, wherein you have 2 translational degrees of freedom, but only one rotational degree of freedom. Within a 4 dimensional space there are 4 orthogonal translations possible, but no less than 6 orthogonal rotations. Or, simpler, when you look at the corner of a room in 3 dimensions, 3 lines meet, and 3 planes meet. But at a corner of a room in 4 dimensions 4 lines and 6 planes meet. Such things are surprises for people trained in thinking in terms of vector algebra. Geometric Algebra makes these kinds of things immediately clear. Also such things like understanding the distinction between ordinary and dual spaces are much clearer to see in Geometric Algebra than in Vector Algebra.

The nice thing about this book in particular is that it shows the most basic of calculations, and writes them out explicitly. This is why I bought the book. This could, in principle, be an ideal preparation for more advanced books in Geometric Algebra.

The reason why I do not give this book more than 3 stars is because the book has some shortcomings and contains errors. The most serious one is in the derivation of the equation m^n= B sin(theta) on page 138 - 139. Because of the misplacement of a minus sign, the whole derivation is clumsy, illogical and, of course, wrong. The final result is true, but the derivation is wrong. The author knew what the result had to be. And, since such an elementary derivation of this result did not exist anywhere else, the author had to find it himself, but made a mistake. That is really a pity, because, since this is the first book on geometric algebra on this elementary level, this particular derivation might give the impression that Geometric Algebra itself is far less elegant than it is.

The mistake the author makes, is thinking that -1-cos2(theta)= - sin2(theta).{With cos2(theta) I mean cos(theta) squared. And with a2 I mean a squared}

So, let me take this opportunity to rectify this, and restore the faith in the elegance of Geometric Algebra.

Begin simply with 1 = mm = mnnm=(mn)(nm)=(m.n + m^n)(n.m + n^m)=(m.n + m^n)(m.n - m^n)=(m.n)2 - (m^n)2.

Therefore (m^n)2=(m.n)2 - 1 = cos2(theta) - 1 = - (1 - cos2(theta))= -sin2(theta). The rest of that derivation then proceeds as in the book.

This derivation is a really nice illustration of the power of Geometric Algebra. By just beginning with the number one, and writing it out in geometric products, the identity the author tries to show, follows.

There is another shortcoming of this book. It does not contain exercises or problems. What I really would like is that every chapter would end with a collection of exercises and problems, with all solutions or the odd numbered solutions in an appendix. Just explaining various results of Geometric Algebra, and showing the details of how to work them out, is not enough to learn the subject, even at this elementary level. Especially with such a novel thing like Geometric Algebra it is essential that you 'get your own hands dirty', and experience yourself solving a problem that would be difficult to do with the old method. Mathematics in particular can ONLY be learned by actually DOING it, through actual problem solving. And the power of a new method only becomes clear by experiencing that you, yourself, are solving a problem that would be difficult, if not impossible without this method.

Since it is the only book that teaches Geometric Algebra on this level I know of, I recommend it, mainly because of a lack of alternatives. That is a negative reason. A positive reason is, that it really succeeds in showing the power of having a COMPLETE algebra of geometric objects at your disposal. This book shows clearly, that very many derivations that are clumsy and difficult to make in vector algebra, become easy and elegant in Geometric Algebra.

Consider also this. Before Newton and Leibniz developed integral calculus, people had to remember formulas of areas and volumes of spheres, cylinders etc. By just learning integral calculus there was no need any more to remember them. You could find many such formulas with only a few strokes of the pen. Moreover, integral calculus made it possible to find formulas for areas and volumes etc. that were not known, and were impossible to find.

In exactly the same way, Geometric Algebra replaces the need for remembering many theorems of geometry. On top of that, it gives a tool to find many geometric theorems that are virtually impossible to find with vectors, and vector algebra. Moreover, it transforms clumsy derivations in other fields into elegant one's. The one example that comes to my mind is the representation of 3 dimensions in quantum mechanics by Pauli matrices or Dirac matrices, which are, according to Geometric Algebra, much better seen as bivectors, trivectors etc. This book gives a very elementary introduction to just that, without being bogged down with details of physics.