The Crust and the β-Skeleton: Combinatorial Curve Reconstruction

Abstract

We construct a graph on a planar point set, which captures its shape in the following sense: if a smooth curve is sampled densely enough, the graph on the samples is a polygonalization of the curve, with no extraneous edges. The required sampling density varies with thelocal feature sizeon the curve, so that areas of less detail can be sampled less densely. We give two different graphs that, in this sense, reconstruct smooth curves: a simple new construction which we call thecrust, and the β-skeleton, using a specific value of β.