On Computing the Fréchet Distance Between Surfaces.

Abstract

We describe two (1+epsilon)-approximation algorithms for computing the Frechet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Frechet distance delta. (1) A 2^{O((n + ( (Area(R)+Area(S))/(epsilon.delta)^2 )^2 )} time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). (2) An O(D/(epsilon.delta)^2) n + 2^{O(D^4/(epsilon^4.delta^2))} time algorithm if R and S are polyhedral terrains over [0,1]^2 with slope at most D. Although, the Frechet distance between curves has been studied extensively, very little is known for surfaces. Our results are the first algorithms (both for surfaces and terrains) that are guaranteed to terminate in finite time. Our latter result, in particular, implies a linear time algorithm for terrains of constant maximum slope and constant Frechet distance.