Matching polyhedral terrains using overlays of envelopes (extended abstract)

Abstract

We show that the combinatorial complexity of the overlay of the lower envelopes of two collections of d-variate piecewise linear functions of overall combinatorial complexity n is Ω(ndα2(n)) and O(nd+ε) for any ε > 0 when d ≥ 2, and O(n2α(n) log n) when d = 2. This extends and improves the analysis of de Berg et al. [9]. We also describe an algorithm that constructs the overlay in the same time. We apply these results to obtain efficient general solutions to the problem of matching two polyhedral terrains in ℝd+1 under translation. For the perpendicular distance measure, which we adopt from functional analysis, we present a matching algorithm that runs in time O(n2d+ε) for any ε > 0. For the directed and undirected Hausdorff distance measures, we present a matching algorithm that runs in time O(nd2+d+ε) for any ε > 0. © Springer-Verlag Berlin Heidelberg 2004.