Osaka Journal of Mathematics

On the flat geometry of the cuspidal edge

Raúl Oset Sinha and Farid Tari

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We study the geometry of the cuspidal edge $M$ in $\mathbb{R}^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the direction of projection varies locally in $S^2$. We also relate the singularities of the height functions and of the projections to some geometric invariants of the cuspidal edge.

Article information

Osaka J. Math., Volume 55, Number 3 (2018), 393-421.

First available in Project Euclid: 4 July 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings 53A05: Surfaces in Euclidean space


Oset Sinha, Raúl; Tari, Farid. On the flat geometry of the cuspidal edge. Osaka J. Math. 55 (2018), no. 3, 393--421.

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