Mathematics

The apparent contour of a surface S in R^3, is the set of critical values of the projection of S on a plane. The apparent contour is a collection of plane 'fronts', i.e. closed curves with double points and cusps. The convex envelope of an apparent contour is the boundary of its convex hull. We classify all singularities up to codimension 3, i.e. all singularities in generic 3-parametric families of convex envelopes of apparent contours. The study was motivated by phase transitions in thermodynamics but the problem is also intimately related to plane nonlinear optimal control problems.