Geodesic
Geometric algebra and singularities of ruled and developable surfaces
Journal of Singularities, Tome 21 (2020), pp. 249-267

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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.
DOI : 10.5427/jsing.2020.21o
Classification : 53A25, 53A05, 15A66, 57R45, 58K40 
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     author = {Junki Tanaka and Toru Ohmoto},
     title = {Geometric algebra and singularities of ruled and developable surfaces},
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Junki Tanaka; Toru Ohmoto. Geometric algebra and singularities of ruled and developable surfaces. Journal of Singularities, Tome 21 (2020), pp. 249-267. doi: 10.5427/jsing.2020.21o

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