Duality of singularities for flat surfaces in Euclidean space

Atsufumi Honda

Journal of Singularities
volume 21 (2020), 132-148

Received: 30 April 2018. Received in revised form: 25 August 2018.

DOI: 10.5427/jsing.2020.21h

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Abstract:

In this paper, we shall discuss the duality of singularities for a class of flat surfaces in Euclidean space. After introducing the definition of the conjugate of a tangent developable, we show that, if a tangent developable admits a swallowtail, its conjugate has a cuspidal cross cap. Similarly, we prove that the conjugate of a tangent developable having cuspidal S^+_1 singularities has cuspidal butterflies, and that cuspidal beaks have self-duality. We also show that cuspidal edges do not possess such a property, by exhibiting an example of a tangent developable with cuspidal edges whose conjugate has 5/2-cuspidal edges. Finally, we prove that conjugates of complete flat fronts with embedded ends cannot be complete flat fronts.

2010 Mathematical Subject Classification:

Primary 53C42; Secondary 57R45.

Key words and phrases:

flat surface, flat front, frontal, singular point, duality of singularities

Author(s) information:

Atsufumi Honda
Department of Applied Mathematics
Faculty of Engineering
Yokohama National University
79-5 Tokiwadai, Hodogaya
Yokohama 240-8501, Japan
email: honda-atsufumi-kp@ynu.ac.jp