Computing large planar regions in termains, with an application to fracture surfaces

We consider the problem of computing the largest region in a terrain that is approximately contained in some two-dimensional plane. We reduce this problem to the following one. Given an embedding of a degree-3 graph G on the unit sphere S-2, whose vertices are weighted, compute a connected subgraph of maximum weight that is contained in some spherical disk of a fixed radius. We give an algorithm that solves this problem in O(n(2) log n(log log n)(3)) time, where it denotes the number of vertices of G or, alternatively, the number of faces of the terrain. We also give a heuristic that can be used to compute sufficiently large regions in a terrain that are approximately planar. We discuss an implementation of this heuristic, and show some experimental results for terrains representing three-dimensional (topographical) images of fracture surfaces of metals obtained by confocal laser scanning microscopy. (C) 2003 Elsevier B.V. All rights reserved.