Vertices and inflexions of plane sections of surfaces in R^3

Abstract

We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. More precisely we will trace the patterns of inflexions and vertices (maxima and minima of curvature) on the sections of a surface as the section passes through a tangential point. We also keep track on the evolution of the curvature of the curves at vertices and control its limit when the vertices collapse at singular points. In particular, we cover all the generic cases, namely when the tangential points are elliptic (A1), umbilic (A1), hyperbolic(A1), parabolic (A2) and cusp of Gauss (A3) points. This has some applications in Computer vision and is also related to interesting mathematical problems such as Legendrian collapse, foliations of surfaces, the 4-vertex Theorem or in general the behaviour of vertices and inflexions in parameter families of curves etc.