Provably good sampling and meshing of surfaces
The notion of ?-sample, introduced by Amenta and Bern, has proven to be a key concept in the theory of sampled surfaces. Of particular interest is the fact that, if E is an ?-sample of a C2-continuous surface S for a sufficiently small ?, then the Delaunay triangulation of E restricted to S is a good approximation of S, both in a topological and in a geometric sense. Hence, if one can construct an ?-sample, one also gets a good approximation of the surface. Moreover, correct reconstruction is ensured by various algorithms. In this paper, we introduce the notion of loose ?-sample. We show that the set of loose ?-samples contains and is asymptotically identical to the set of ?-samples. The main advantage of loose ?-samples over ?-samples is that they are easier to check and to construct. We also present a simple algorithm that constructs provably good surface samples and meshes. Given a C2-continuous surface S without boundary, the algorithm generates a sparse ?-sample E and at the same time a triangulated surface Del|S(E). The triangulated surface has the same topological type as S, is close to S for the Hausdorff distance and can provide good approximations of normals, areas and curvatures. A notable feature of the algorithm is that the surface needs only to be known through an oracle that, given a line segment, detects whether the segment intersects the surface and, in the affirmative, returns the intersection points. This makes the algorithm useful in a wide variety of contexts and for a large class of surfaces.
?-sampling surface mesh generation surface approximation restricted delaunay triangulation mesh refinement