Reconstructing a Polyhedron between Polygons in Parallel Slices

Abstract

Given two $n$-vertex polygons, $P=(p_1, \ldots, p_n)$ lying in the $xy$-plane at $z=0$, and $P'=(p'_1, \ldots, p'_n)$ 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 $\pi_i$ connecting $p_i$ to $p'_i$ for all $i =1, \ldots, n$. The surface then consists of bands, where the $i$th band goes between $\pi_i$ and $\pi_{i+1}$. 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 $i$th 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.