https://hal.archives-ouvertes.fr/hal-00488829Boissonnat, Jean-DanielJean-DanielBoissonnatGEOMETRICA - Geometric computing - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en AutomatiqueOudot, SteveSteveOudotGEOMETRICA - Geometric computing - CRISAM - Inria Sophia Antipolis - Méditerranée - Inria - Institut National de Recherche en Informatique et en Automatique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en AutomatiqueProvably good sampling and meshing of surfacesHAL CCSD2005Computational geometrycomputational topologymesh generationsurface approximation[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]Boissonnat, Jean-DanielIST Programme of the EU as a Shared-cost RTD (FET Open) Project under Contract No IST-2000-26473 (EC - INCOMING - 2010-06-03 09:53:432022-01-20 17:28:212010-06-03 10:13:10enJournal articlesapplication/pdf1The notion of e-sample, as 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 e-sample of a smooth surface S for a sufficiently small e, 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 e-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 e-sample. We show that the set of loose e-samples contains and is asymptotically identical to the set of e-samples. The main advantage of loose e-samples over e-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 smooth surface S without boundary, the algorithm generates a sparse e-sample E and at the same time a triangulated surface. 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 remarkable 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.