Reconstructing a Polyhedron between Polygons in Parallel Slices

• Published in: arXiv.org
• Main Authors: Biedl, Therese, Bulatovic, Pavle, Irvine, Veronika, Lubiw, Anna, Merkel, Owen, Anurag Murty Naredla
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 13.04.2020
• Subjects: Algorithms; Polygons; Polynomials; Triangulation
• ISSN: 2331-8422

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.