Duality on generalized cuspidal edges preserving singular set images and first fundamental forms

Atsufumi Honda, Kosuke Naokawa, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada

Journal of Singularities
volume 22 (2020), 59-91

Received: 23 February 2019. Received in revised form: 29 May 2020.

DOI: 10.5427/jsing.2020.22e

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Abstract:

In the second, fourth and fifth authors' previous work, a duality on generic real analytic cuspidal edges in the Euclidean 3-space R^3 preserving their singular set images and first fundamental forms, was given. Here, we call this an "isometric duality". When the singular set image has no symmetries and does not lie in a plane, the dual cuspidal edge is not congruent to the original one. In this paper, we show that this duality extends to generalized cuspidal edges in R^3, including cuspidal cross caps, and 5/2-cuspidal edges. Moreover, we give several new geometric insights on this duality.

2010 Mathematical Subject Classification:

57R45; 53A05

Key words and phrases:

singularity, wave front, cuspidal edge, first fundamental form

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

Kosuke Naokawa
Department of Computer Science
Faculty of Applied Information Science
Hiroshima Institute of Technology
2-1-1 Miyake, Saeki, Hiroshima, 731-5193, Japan
email: k.naokawa.ec@cc.it-hiroshima.ac.jp

Kentaro Saji
Department of Mathematics
Faculty of Science
Kobe University
Rokko, Kobe 657-8501
email: saji@math.kobe-u.ac.jp

Masaaki Umehara
Department of Mathematical and Computing Sciences
Tokyo Institute of Technology
Tokyo 152-8552, Japan
email: umehara@is.titech.ac.jp

Kotaro Yamada
Department of Mathematics
Tokyo Institute of Technology
Tokyo 152-8551, Japan
email: kotaro@math.titech.ac.jp