Geometric algebra and singularities of ruled and developable surfaces

Junki Tanaka and Toru Ohmoto

Journal of Singularities
volume 21 (2020), 249-267

Received: 24 March 2018. Received in revised form: 7 August 2018.

DOI: 10.5427/jsing.2020.21o

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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^\∞ 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 τ^, 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.

2010 Mathematical Subject Classification:

53A25, 53A05, 15A66, 57R45, 58K40

Key words and phrases:

Differential line geometry, Clifford algebra, Ruled surfaces, Developable surfaces, Singularities of smooth maps

\address{T.~Ohmoto, Department of Mathematics, Faculty of Science, Global Station of Bigdata and cybersecurity (GSB), Hokkaido University, Sapporo 060-0810, Japan}

Author(s) information:

Junki Tanaka	Toru Ohmoto
Kobo Co, LTD	Department of Mathematics
Kobe, Japan	Faculty of Science
	Global Station of Bigdata
	and cybersecurity
	Hokkaido University
	Sapporo 060-0810, Japan
email: junki.tanaka@khobho.co.jp	email: ohmoto@math.sci.hokudai.ac.jp