- Gebundene Ausgabe: 530 Seiten
- Verlag: Morgan Kaufmann (6. Februar 2006)
- Sprache: Englisch
- ISBN-10: 0120884003
- ISBN-13: 978-0120884001
- Größe und/oder Gewicht: 19,8 x 2,9 x 24 cm
- Durchschnittliche Kundenbewertung: Schreiben Sie die erste Bewertung
- Amazon Bestseller-Rang: Nr. 324.692 in Fremdsprachige Bücher (Siehe Top 100 in Fremdsprachige Bücher)
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Visualizing Quaternions (The Morgan Kaufmann Series in Interactive 3D Technology) (Englisch) Gebundene Ausgabe – 6. Februar 2006
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"This is exactly the caliber of book needed. A lot of professionals will benefit from this one." --Dave Eberly, Geometric Tools, Inc.
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The book also serves as an accessible introduction into moving frames, which are the "right" way of treating the "extrinsic" differential geometry of curves and surfaces and which benefit a lot from a treatment using quaternions and spinors, leading to simpler equations. Some of the ideas presented here have been developed into a full-blown quaternionic framework for "extrinsic" surface geometry, by mathematicians of the Berlin school (U. Pinkall, A. Bobenko, etc.). The Caltech thesis of K. Crane ("Conformal Geometry Processing") might serve as a natural follow-up reading, for those whose interest has been piqued by Hanson's book.
It is also appreciated, that Hanson emphasizes the "square root" nature of quaternions, that's key to understand physical concepts like spin, the Dirac equation and supersymmetry, which are of tremendous importance in geometry and topology as well as theoretical physics, but may appear bizarre and abstract without some geometric intuition.
Part 1 is an introduction for those readers new to the topic. As far as introductions go, it is not too bad. It does in fact contain one important subject - quaternion interpolation - that is not always covered in other texts. Hanson covers interpolation in part 1 and again in part 2. If your interest is computer animation, this may be sufficient reason to acquire the book...analogous to purchasing an album just to get one song. However, if you are completely new to quaternions and want to develop a firm intuition grounded in first principles, then a book that is at least an order of magnitude better is "Quaternions and Rotation Sequences" by J. B. Kuipers.
Parts 2 and 3 are the most interesting parts of the book. Hanson presents a series of small chapters that discuss quaternions from different advanced mathematical viewpoints (differential geometry, group theory, Clifford algebras, octonions). The chapters are small, and so they by necessity contain references to the literature where the considerable background material required for understanding the topics is developed. If you have a good background in differential geometry and some abstract algebra, then the chapters are quite nice. In this sense, parts 2 and 3 of the book are more appropriate for mathematicians.
The technique of including routine, "turn the crank" type of calculations in the text, and deferring the sometimes considerable details and theory to references allows Hanson to cover more topics than usual. However, it is exactly those details that distinguish between what is useful and well conceived mathematical theory from mathematical gibberish. Deferring details to the literature can also encourage an over-reaching of the author beyond his understanding of the material. Hanson has walked a fine line here, but still I must mention two issues that I found annoying:
1) A Riemannian manifold is not specified only by giving the charts ("local patches") as Hanson seems to think on page 352. One must also add constraints on the topology -- typically Hausdorff with a countable basis of open sets. These are not just moot considerations; the topology allows a construction of a partition of unity which in turns guarantees the existence of the Riemannian metric. In particular, the mild condition of paracompactness will ensure the existence of the partition of unity.
2) It is a gross over-simplification, and mathematically non-trivial, to claim the basis vectors of Euclidean space have precise analogs in Fourier transform theory, as Hanson does on page 340. Heuristic analogs...yes... but precise analogs?...only if one has developed the necessary mathematical machinery using the theory of distributions. The inner product relation ei.ej = kronecker delta ij given by Hanson on page 340 would have to be generalized to a delta function. It was one of the major accomplishments of 20th century mathematics that Schwartz was able to put the delta function on a firm mathematical basis with his theory of distributions (for which he received the Fields medal) Before Schwartz, delta functions were at best a useful computational tool in the hands of physicists like Dirac who were guided by their physical intuition, and at worst, an example of the mathematical gibberish alluded to earlier.
In short, this is a good book for those with the mathematical prerequisites. Those with a more traditional background in computer science might be advised to first peruse a copy at their local bookstore to verify it matches their interests.