Graceful planes and lines

Abstract

In the framework of the theory of arithmetic geometry (Reveillès, Géométrie Discrète, Calcul en Nombres Entiers et Algorithmique, Thèse d’état, Université Louis Pasteur, Strasbourg, 1991), we propose an approach to discretize polyhedra by meshes of discrete triangles. We propose a general discretization scheme based on reducing the 3D problem to a 2D problem. We introduce new classes of discrete planes and lines called graceful planes and graceful lines. Naive planes and graceful lines are used to construct as thin as possible triangular mesh discretization admitting an analytical description. The interiors of the triangles are portions of naive planes, while the sides are graceful lines. These primitives serve as an optimal ground for obtaining thin tunnel-free discretizations, within the adopted generation scheme. We also extend our considerations to arbitrary surfaces and curves.