Aspects of Mesh Theory

Abstract

The main purpose of this paper is to give a thorough account of planar quadrilateral and offset meshes. To this end, abstract meshes, the linear spaces C(M), P(M), and the distance functions d*(M1 , M2 ) are defined and applied to planar quadrilateral meshes. We then study the discrete analogue of the Gauss map as well as the defining properties of circular and conical meshes. The proof of the angle condition for conical meshes is given and used as motivation for the study of Lie sphere geometry. We apply the theory of Laguerre and Moebius transformations to conical and circular meshes respectively. All of this theory is then applied to the creation of a planar quadrilateral mesh with the face offset property in the context of a study project.