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We investigate the vertex curve, that is the set of points in the hyperbolicregion of a smooth surface in real 3-space at which there is a circle in thetangent plane having at least 5-point contact with the surface. The vertexcurve is related to the differential geometry of planar sections of the surfaceparallel to and close to the tangent planes, and to the symmetry sets ofisophote curves, that is level sets of intensity in a 2-dimensional image. Weinvestigate also the relationship of the vertex curve with the parabolic andflecnodal curves, and the evolution of the vertex curve in a generic1-parameter family of smooth surfaces.

Contact with circles and Euclidean invariants of smooth surfaces in R^3