  Mark all vertices on a curve evolving under a family of curves obtained by
intersecting a smooth surface M with the 1-parameter family of planes parallel
to the tangent plane to M at a point p. Those vertices trace out a set, called
the vertex set of M through p. We take p to be an isolated umbilic point on M
and describe the perestroikas of the vertex set under generic n-parameter small
deformations of the surface, as well as the corresponding discriminants. One of
the main consequences of our results is that, in some sense (see Theorem 2),
generically the study of the discriminants of small n-deformations, with n
greater than or equal to 2, simplifies to that of 2-deformations.
  This work was primarily motivated by the medial representation of shapes in
Computer Vision and Image Analysis, where the behaviour of vertices plays a
crucial role in the qualitative changes of the skeleton or Blum medial axis of
curves.
  On the other hand, the study of vertices of curves has raised up a great
interest in particular in Geometry and Singularity Theory, in line with several
problems such as the 4-vertex Theorem, the local geometry of surfaces and
Geometry of Caustics. These classical subjects have received a new impulsion
due to the development of Contact Geometry, especially in the works of V.
Arnold on Legendrian Collapse and Legendrian Sturm Theory showing the relation
between vertices of plane curves and Legendrian singularities and Sturm-Hurwitz
Theorem.
