Topologically consistent algorithms related to convex polyhedra

Abstract

The paper presents a general method for the design of numerically robust and topologically consistent geometric algorithms concerning convex polyhedrain the three-dimensional space. A graph is the vertex-edge graph of a convex polyhedron if and only if it is planar and triply connected (Steinitz’ theorem). On the basis of this theorem, conventional geometric algorithms are revised in such a way that, no matter how poor the precision in numerical computation may be, the output graph is at least planar and triply connected. The resultant algorithms are robust in the sense that they do not fail in finiteprecision arithmetic, and are consistent in the sense that the output is the true solution to a perturbed input.