The geometry of conflict sets

Abstract

A conflict set measures where distance functions from different submanifolds in one ambient manifold have critical points with a coinciding critical value. This critical value is not necessarily a minimum. The conflict set is a variation on the concept of a medial axis. Chapters one and two of the underlying book are concerned with generalizations of the so-called mirror beam equation of geometrical optics. Amongst others a torsion formula for conflict sets is derived in chapter two. In chapters three and four systematic use is made of canonical or symplectic relationss to show that generically conflict sets lift to Legendrian submanifolds of the spherical cotangent bundle of the ambient manifold. Numerous other geometrical constructions, of the pedal curve for instance, are also treated using the symplectic framework. In chapter five the machinery of singularity theory is used to show that generic singularities of conflict sets are singularities occuring in families of wavefronts.