Flat surfaces along cuspidal edges

Shyuichi Izumiya, Kentaro Saji, and Nobuko Takeuchi

Journal of Singularities
volume 16 (2017), 73-100

Received: 23 May 2016. Received in revised form: 8 May 2017.

DOI: 10.5427/jsing.2017.16c

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We consider developable surfaces along the singular set of a cuspidal edge surface which are regarded as flat approximations of the cuspidal edge surface. For the study of singularities of such developable surfaces, we introduce the notion of Darboux frames along cuspidal edges, and introduce invariants. As a by-product, we introduce the notion of higher-order helices which are generalizations of previous notions of generalized helices (i.e., slant helices and clad helices). We use this notion to characterize special cuspidal edges.


cuspidal edges, flat approximations, curves on surfaces, Darboux frame, developable surfaces, slant helices, clad helices, k-th-order helices, contour edges, isophotic edges

Mathematical Subject Classification (2010):

Primary 57R45; Secondary 58Kxx

Author(s) information:

Shyuichi Izumiya Kentaro Saji Nobuko Takeuchi
Department of Mathematics Department of Mathematics Department of Mathematics
Hokkaido University Kobe University, Rokko 1-1 Tokyo Gakugei University, Koganei
Sapporo 060-0810, Japan Nada, Kobe 657-8501, Japan Tokyo, 184-8501, Japan
email: izumiya@math.sci.hokudai.ac.jp email: saji@math.kobe-u.ac.jp email: nobuko@u-gakugei.ac.jp