Parabolic curves of evolving surfaces

In this article we show how certain geometric structures which are also associated with a smooth surface evolve as the shape of the surface changes in a 1-parameter family. We concentrate on the parabolic set and its image under the Gauss map, but the same techniques also classify the changes in the dual of the surface. All these have significance for computer vision, for example through their connection with specularities and apparent contours. With the aid of our complete classification, which includes all the phenomena associated with multi-contact tangent planes as well as those associated with parabolic sets, we re-examine examples given by J. Koenderink in his book (1990) under the title of Morphological Scripts. We also explain some of the connections between parabolic sets and ridges of a surface, where principal curvatures achieve turning values along lines of curvature. The point of view taken is the analysis of the contact between surfaces and their tangent planes. A systematic investigation of this yields the results using singularity theory. The mathematical details are suppressed here and appear in Bruce et al. (1993).