Reconstructing a Polyhedron between Polygons in Parallel Slices
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Given two n-vertex polygons, P=(p_1, …, p_n) lying in the xy-plane at z=0, and P'=(p'_1, …, p'_n) lying in the xy-plane at z=1, a banded surface is a triangulated surface homeomorphic to an annulus connecting P and P' such that the triangulation's edge set contains vertex disjoint paths π_i connecting p_i to p'_i for all i =1, …, n. The surface then consists of bands, where the ith band goes between π_i and π_i+1. We give a polynomial-time algorithm to find a banded surface without Steiner points if one exists. We explore connections between banded surfaces and linear morphs, where time in the morph corresponds to the z direction. In particular, we show that if P and P' are convex and the linear morph from P to P' (which moves the ith vertex on a straight line from p_i to p'_i) remains planar at all times, then there is a banded surface without Steiner points.
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