J.-D. Boissonnat, S. Y. Oudot. Provably Good Sampling and Meshing of Lipschitz Surfaces. Proc. 22nd Annual ACM Sympos. Comput. Geom., pages 337-346, 2006.


In the last decade, a great deal of work has been devoted to the elaboration of a sampling theory for smooth surfaces. The goal was to ensure a good reconstruction of a given surface S from a finite subset E of S. The sampling conditions proposed so far offer guarantees provided that E is sufficiently dense with respect to the local feature size of S, which can be true only if S is smooth since the local feature size vanishes at singular points. In this paper, we introduce a new measurable quantity, called the Lipschitz radius, which plays a role similar to that of the local feature size in the smooth setting, but which turns out to be well-defined and positive on a much larger class of shapes. Specifically, it characterizes the class of Lipschitz surfaces, which includes in particular all piecewise smooth surfaces such that the normal variation around singular points is not too large. Our main result is that, if S is a Lipschitz surface and E is a sample of S such that any point p of S is at distance less than a fraction of the Lipschitz radius of S, then we obtain similar guarantees as in the smooth setting. More precisely, we show that the Delaunay triangulation of E restricted to S is a 2-manifold isotopic to S lying at bounded Hausdorff distance from S, provided that its facets are not too skinny. We further extend this result to the case of loose samples. As a straightforward application, the Delaunay refinement algorithm we proved correct for smooth surfaces works fine and offers the same topological and geometric guarantees for Lipschitz surfaces.


, author =      {J.-D. Boissonnat and S. Oudot}
, title =       {Provably Good Sampling and Meshing of {Lipschitz} Surfaces}
, year =        {2006}
, pages =	{337-346}
, booktitle =	{Proc. 22nd Annual ACM Sympos. Comput. Geom.}