Abstract
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.
Citation
Raúl Oset Sinha. Farid Tari. "On the flat geometry of the cuspidal edge." Osaka J. Math. 55 (3) 393 - 421, July 2018.