Orthogonal triangulation of polygons

Abstract

Triangulation of polygons arises in various applications, such as computer graphics or automatic designing systems. One of the goals of triangulation is to replace some, usually convex, polygonal area into fragments, for which the test and design tasks are performed by the most simple and unified way. Most of the known algorithms work with the triangles. The question of whether or not you can always assume that the polygon belongs to a rectangular net and is made up of rectangles and right triangles of the net, perhaps for the first time, was placed by N. Yu. Dodonov in connection with the works for the reconstruction of parametric surfaces specified with tabular data. Despite the fact that the answer to this question is positive for triangles (obviously) and convex quadrilaterals, in this paper it is proved that a 7-dimensional space of convex pentagons set of untriangulated 5-gon has a non-empty interior. Thus, it denied even the possibility of approximation this polygon by triangulated ones. Refs 3. Figs 5. Tables 2.