TY - JOUR

T1 - Geometric algebra and singularities of ruled and developable surfaces

AU - Tanaka, Junki

AU - Ohmoto, Toru

N1 - Funding Information:
This paper is based on the first author’s master thesis [22]. This work was supported by JSPS KAKENHI Grant Numbers JP15K13452, JP17H0612818 and JP18K18714.
Publisher Copyright:
© 2020, Worldwide Center of Mathematics. All rights reserved.

PY - 2020

Y1 - 2020

N2 - Any ruled surface in R3 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 (R2, 0) → (R3, 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.

AB - Any ruled surface in R3 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 (R2, 0) → (R3, 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.

KW - Clifford algebra

KW - Developable surfaces

KW - Differential line geometry

KW - Ruled surfaces

KW - Singularities of smooth maps

UR - http://www.scopus.com/inward/record.url?scp=85082127581&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85082127581&partnerID=8YFLogxK

U2 - 10.5427/jsing.2020.21o

DO - 10.5427/jsing.2020.21o

M3 - Article

AN - SCOPUS:85082127581

VL - 21

SP - 249

EP - 267

JO - Journal of Singularities

JF - Journal of Singularities

SN - 1949-2006

ER -