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Surface evolution and representation using geometric algebra. (English) Zbl 0996.51008

Cipolla, Roberto (ed.) et al., The mathematics of surfaces IX. Proceedings of the 9th IMA conference, Cambridge, GB, September 4-7, 2000. London: Springer. 144-168 (2000).
Summary: This paper uses the mathematical language of geometric algebra based on the algebras of Clifford and Grassmann [see D. Hestenes and G. Sobczyk, ‘Clifford algebra to geometric calculus’, D. Reidel, Dordrecht (1984; Zbl 0541.53059); D. Hestenes, ‘New-foundations for classical mechanics’, 2nd ed., Kluwer Academic Publishers (1999; Zbl 0932.70001)].
A 4d (projective) description of 3d Euclidean space is used extensively in computer version and computer graphics where rotations and translation can be described by a single \(4\times 4\) matrix and nonlinear projective transformations become linear. Projective geometry fits very nicely into the geometric algebra framework.
Recently, the application of the idea in which a 5d conformal space is used as the representation of 3d Euclidean space has been the subject of renewed interest. This paper uses the conformal representation to look at the problems of surfaces representation and evolution, and of wavefront propagation from such surfaces.
For the entire collection see [Zbl 0946.00021].

MSC:

51N25 Analytic geometry with other transformation groups
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
15A15 Determinants, permanents, traces, other special matrix functions

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