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GEOMETRIC ALGEBRA AND SINGULARITIES OF RULED AND DEVELOPABLE SURFACES

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Title: GEOMETRIC ALGEBRA AND SINGULARITIES OF RULED AND DEVELOPABLE SURFACES
Authors: Tanaka, Junki Browse this author
Ohmoto, Toru Browse this author
Keywords: Differential line geometry
Clifford algebra
Ruled surfaces
Developable surfaces
Singularities of smooth maps
Issue Date: 22-Oct-2020
Publisher: Worldwide Center of Mathematics
Journal Title: Journal of Singularities
Volume: 21
Start Page: 249
End Page: 267
Publisher DOI: 10.5427/jsing.2020.21o
Abstract: Any ruled surface in R-3 is described as a curve of unit dual vectors in the algebra of dual quaternions (=the even Clifford algebra Cl+ (0, 3, 1)). Combining this classical framework and A-classification theory of C-infinity map-germs (R-2, 0) -> (R-3, 0), we characterize local diffeomorphic types of singular ruled surfaces in terms of geometric invariants. In particular, using a theorem of G. Ishikawa, we show that local topological type of singular developable surfaces is completely determined by vanishing order of the dual torsion tau, that generalizes an old result of D. Mond for tangent developables of non-singular space curves. This work suggests that Geometric Algebra would be useful for studying singularities of geometric objects in classical Klein geometries.
Type: article
URI: http://hdl.handle.net/2115/79607
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

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