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Geometric algebra with applications in engineering. (English) Zbl 1179.15025

Geometry and Computing 4. Berlin: Springer (ISBN 978-3-540-89067-6/hbk). xiv, 385 p. (2009).
Both, the discovery of the quaternions by W. R. Hamilton (1843) and the “Lineale Ausdehnungslehre” by H. G. Grassmann (1844) influenced W. K. Clifford (1873) to the invention of the “Geometric algebras”. It is mainly the credit of D. Hestenes who revitalized these old ideas and initiated new approaches to such modern disciplines as computer vision, camera modeling, monocular pose estimation and robotics.
The author introduces the principle of outer morphisms as generalizations of linear mappings in geometric algebras. New notions like “blades”, “versor”, “joins” are carefully introduced. Duality relations between the inner and outer products in the corresponding geometric algebras are discussed. “Meet” and “Join” operations of two blades are defined. While “blades” are outer products of 1-vectors, versors can be understood as geometric products of 1-vectors which do not vanish. The author makes use from tensor representations in order to simplify the algorithmization of differentiation and integration.
Reflection and rotation are described by explicit formulae within his new terminology. A homogenisation procedure leads to an embedding of the Euclidean space into an affine space. These embeddings are important for visualization problems. In particular the geometric algebras \(G_{n+1,1}\) are studied and geometric objects like points, spheres, titles, lines and pairs of points interpreted. Conic-sections are constructed in suited geometric algebras. The representation theory enables numerical calculations. For instance non-trivial equations of Lyapunov type have to be solved. The notion of a versor appears as advantageously to describe the necessary operations. The author also considers the problem of the error expansion in geometric products. “Uncertainties” caused by fuzzy dates are studied in projective, conformal and conic spaces.
It is of special interest to consider the description of “constraint” metrics and its meaning for the estimation of geometric terms. A new camera model (inversion camera) is presented and detailed discussed. The book contains statements on the advantageous positioning of monoculare cameras. Some new aspects to Farouki’s hodograph curves are presented. These curves are very important for the solution of design problems.
In the future, geometric algebras will open new perspectives for the treatment of multidimensional problems. This book is a very nice, comprehensive and methodically perfected presentation of a new important branch of mathematics. The book shows there is no separation line between theoretical and applied mathematics. Mathematics has to be understood as a monolithic whole – this is one of the messages of this warmly recommended monography.

MSC:

15A66 Clifford algebras, spinors
51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
68-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science
70E60 Robot dynamics and control of rigid bodies
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
65D19 Computational issues in computer and robotic vision
15-02 Research exposition (monographs, survey articles) pertaining to linear algebra
51-04 Software, source code, etc. for problems pertaining to geometry

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