Faceting of curved surfaces using the curvature coordinate system

In many situations, a curved surface has to be approximated by a facetted surface, i.e., as a network with planar meshes. Most often this is done by triangulation of the surface. Points are chosen on the surface and the points are connected by straight lines so that these lines make a network of triangular meshes and so that no normal to the curved surface intersect more than one mesh/facet. The result is a faceted surface, with vertices, edges and triangular facets. But faceting a curved surface can also be done using planes as the basic geometrical element instead of points. One way of doing this is by tangent faceting. Tangent points are chosen on the surface and the tangent planes at these points are connected along lines of intersection so that these lines make a network with planar meshes and so that no normal to the curved surface intersect more than one facet. The result is a faceted surface with facets, edges and, unless special effort are made, three-way vertices. On facetted surfaces the Gaussian curvature is concentrated at the vertices. For triangular faceted surfaces the sign of the Gaussian curvature is often seen to differ locally from the sign of the Gaussian curvature of the curved surface. If the curvature coordinate system [1] is used in the design process of triangulated as well as tangent-faceted surfaces, both the topology of the system and the sign of the Gaussian curvature of the facetted surface can be determined deliberately by the designer. This paper describes the faceting processes and the geometric rules that determine the topology and the curvature of the facetted surface when given form in the curvature coordinate system.