Thin discrete triangular meshes

In this paper we present an approach to describe polyhedra by meshes of dis crete triangles. The study is based on the theory of arithmetic discrete ge ometry (J.-P. Reveilles, Geometrie discrete, calcul en nombres entiers ct a lgorithmique, These d'etat, Universite Louis Pasteur, Strasbourg, December 1991). As distinct from the previous investigations on this topic, the tria ngles we introduce are parts of the thinnest possible discrete 6-tunnel-fre e planes, i.e., those that are usually used in practice. Given a plane P in the space, we define a 6-tunnel-free discrete plane, cal led a regular plane, which appears to be the best approximation to P. Given a mesh of triangles, we propose a method to approximate any triangle by a discrete triangular patch - a portion of a regular plane, and we prove that the resulting triangular mesh is 6-tunnel-free. The properties of the appr oximation obtained make the suggested approach convenient for practical app lications. (C) 2000 Elsevier Science B.V. All rights reserved.