On Computing the Fréchet Distance Between Surfaces

We describe two (1 + ε)-approximation algorithms for computing the Fréchet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Fréchet distance δ. 1. A 2 (( n+ Area(R)+Area(S) (εδ)2 )2) time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). 2. An O(D/(εδ)2) · n + 2O(D/(εδ)) time algorithm if R and S are polyhedral terrains over [0, 1]2 with slope at most D. Although, the Fréchet 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 Fréchet distance. 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes, F.2.2 Nonnumerical Algorithms and Problems