The curvature coordinate system

The paper describes a concept for a curvature coordinate system on regular curved surfaces from which faceted surfaces with plane quadrangular facets can be designed. The lines of curvature are used as parametric lines for the curvature coordinate system on the surface. A new conjugate set of lines, called middle curvature lines, is introduced. These lines define the curvature coordinate system. Using the curvature coordinate system, the surface can be conformally mapped on the plane. In this mapping, elliptic sections are mapped as circles, and hyperbolic sections are mapped as equilateral hyperbolas. This means that when a plane orthogonal system of curves for which the vertices in a mesh always lie on a circle is mapped on a surface with positive Gaussian curvature using inverse mapping, and the mapped vertices are connected by straight lines, this network will form a faceted surface with plane quadrangular facets. Examples of such faceted surfaces are shown. For surfaces with negative Gaussian curvature, one type of curve system that forms faceted surfaces with plane quadrangular facets has been found.