  We investigate the vertex curve, that is the set of points in the hyperbolic
region of a smooth surface in real 3-space at which there is a circle in the
tangent plane having at least 5-point contact with the surface. The vertex
curve is related to the differential geometry of planar sections of the surface
parallel to and close to the tangent planes, and to the symmetry sets of
isophote curves, that is level sets of intensity in a 2-dimensional image. We
investigate also the relationship of the vertex curve with the parabolic and
flecnodal curves, and the evolution of the vertex curve in a generic
1-parameter family of smooth surfaces.
