arXiv:math/0502552v1 [math.DG] 25 Feb 2005 Vertices and inflexions of plane sections of smooth surfaces in R 3 Andr´ e Diatta and Peter Giblin University of Liverpool, Liverpool L69 3BX, England email adiatta@liv.ac.uk pjgiblin@liv.ac.uk Abstract We discuss the behaviour of vertices and inflexions on one-parameter families of plane curves, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a smooth surface by parallel planes. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere. 1 Introduction Let M be a smooth surface, and p be a point of M . We shall consider the intersection of M with a family of planes parallel to the tangent plane at p. This family of plane curves contains a singular member, when the plane is the tangent plane itself; generically the other members of the family close to the tangent plane are nonsingular curves. The motivation for this work comes from computer vision, where the surface is the intensity surface z = f (x, y) corresponding to the intensity function f of a two-dimensional image, and the plane curves are level sets of this function, that is isophotes. A great deal of information about the shape of these level sets and the way they evolve through the singular level set is contained in the family of so-called symmetry sets and medial axes of the level sets (see for example [8]). These sets in turn take some of their structure from the pattern of vertices and inflexions (curvature extrema and zeros) of the level set. In this article we concentrate on the vertices and inflexions, and apply this and other results to the study of symmetry sets in articles to appear elsewhere [6, 7]. Besides the patterns of vertices and inflexions we also study the limiting curvatures at the vertices as the level set approaches the singular member of the family. The contact between a surface and its tangent plane at p is an affine invariant of the surface. Likewise the inflexions on the intersections with nearby planes are affine invariants, but we are also interested in the curvature extrema on these sections, and these are euclidean invariants. For a generic surface M , the contact between the surface and its tangent plane at a point p, as measured by the height function in the normal direction at p, can be of the following types. See for example [10] for the geometry of these situations, and [4, 5, 9] for an extensive discussion of the singularity theory. • The contact at p is ordinary (‘A 1 contact’), at an elliptic point or at a hyperbolic point (occupying regions of M ). The intersection of M with its tangent plane at p is locally an isolated point or a pair of transverse smooth arcs. (From the point of view of contact there is no distinction between ‘ordinary’ elliptic points and umbilics, where the principal curvatures coincide. But as we shall see there is a great deal of difference when we consider vertices of the plane sections.) 1