ON THE MATABILITY OF POLYGONS

Abstract

Interpolating a piecewise-linear triangulated surface between two polygons lying in parallel planes has attracted a lot of attention in the literature over the past three decades. This problem is the simplest variant of interpolation between parallel slices, which may contain multiple polygons with unrestricted geometries and/or topologies. Its solution has important applications to medical imaging, digitization of objects, geographical information systems, and more. Practically all currently-known methods for surface reconstruction from parallel slices assume a priori the existence of a non-self-intersecting triangulated surface defined over the vertices of the two polygons, which connects them. Gitlin et al. were the first to specify a nonmatable pair of polygons. In this paper we provide proof of the nonmatability of a “simpler” pair of polygons, which is less complex than the example given by Gitlin et al. Furthermore, we provide a family of polygon pairs with unbounded complexity, which we believe to be nonmatable. We also give a few sufficient conditions for polygon matability.