Geodesic complexity of motion planning

Abstract

We introduce the geodesic complexity of a metric space, inspired by the topological complexity of a topological space. Both of them are numerical invariants, but, while the TC only depends on the homotopy type, the GC is an invariant under isometries. We show that in many cases they coincide but we also develop tools to distinguish the two in a range of examples. To this end, we study what we denote the total cut locus, which does not appear to have been explicitly considered in the literature. To the knowledge of the author, the GC is a new invariant of a metric space. Furthermore, just like the TC, the GC has potential applications to the field of robotics.