“Shortest” arcs in closed planar disks vary continuously with the boundary

Given a triple (D,a,b) where D a closed topological disk in the plane with distinct points {a,b}⊂D there is not necessarily a path α of finite length connecting a to b in D. Nevertheless the notion of shortest arc extends naturally to the topological category as follows. There exists a continuous operator geo which assigns to each triple (D,a,b) a closed arc α⊂D such that α connects a and b, and moreover α has uniquely minimal finite Euclidean arclength whenever possible.