Surface reconstruction between simple polygons via angle criteria

Abstract

We consider the problem of connecting two simple polygons P and Q in parallel planes by a polyhedral surface. The goal is to find an optimality criterion, which naturally satifies the following conditions: (i) if P and Q are convex, then the optimal surface is the convex hull of P and Q (without facets P and Q), and (ii) if P can be obtained from Q by scaling with a center c, then the optimal surface is the portion of the cone defined by P and apex c between the two planes. We provide a criterion (based on the sequences of angles of the edges of P and Q) which satisfies these conditions, and for which the optimal surface can be efficiently computed. Moreover, we supply a condition, so-called angle consistency, which proved very helpful in preventing self intersections (for our and other criteria). The methods have been implemented and gave improved results in a number of examples.