The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.

Given a family P fP 1;...; P mg of m polygonal regions (possibly intersecting) in the plane, we consider the problem of locating a straight line ‘ intersecting the convex hull of P and such that minkd(Pk,‘) is maximal. We give an algorithm that solves the problem in time O((m 2 + nlogm)logn) using O(m 2 + n) space, where n is the total number of vertices of P1,...,Pm. The previous best running time for this problem was O(n 2) [J. Janardan, F.P. Preparata, Widest-corridor problems, Nordic Journal of Computing 1 (1994) 231–245]. We also improve the known complexity for several variants of this problem which include a three dimensional version – the maximin plane problem –, the weighted problem and considering measuring distance different to the Euclidean one.