On the flat geometry of the cuspidal edge (Q1667184)

The authors study the geometry of the cuspidal edge \(M\) in \(\mathbb{R}^3\) derived from its contact with planes and lines. The contact of \(M\) with planes is measured by the singularities of the height functions on \(M\). They 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 \(M\). The authors list the generic singularities of the projections and obtain the generic deformations of the apparent contour when the direction of projection varies locally in \(S^2\). They also relate the singularities of the height functions and of the projections to some geometric invariants of the cuspidal edge. The paper is organized into four sections as follows: Introduction, Preliminaries (geometric cuspidal edge, classification tools), Functions on a cuspidal edge (the geometry of functions on a cuspidal edge, contact of a geometric cuspidal edge with planes), Orthogonal projections of a cuspidal edge (apparent contour of a cuspidal edge). Another paper of the second author directly connected to this subject is [\textit{F. Tari}, J. Lond. Math. Soc., II. Ser. 44, No. 1, 155--172 (1991; Zbl 0767.53004)].