Edge-Unfolding Nearly Flat Convex Caps
Abstract
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in R^3 has an edge-unfolding to a non-overlapping polygon in the plane. A convex cap is the intersection of the surface of a convex polyhedron and a halfspace. "Nearly flat" means that every outer face normal forms a sufficiently small angle phi < Phi with the z-axis orthogonal to the halfspace bounding plane. The size of Phi depends on the acuteness gap alpha: if every triangle angle is at most pi/2-alpha, then Phi ~= 0.36 sqrt(alpha) suffices; e.g., for alpha ~= 3deg, Phi = 5deg. Even if C is closed to a polyhedron by adding the convex polygonal base under C, this polyhedron can be edge-unfolded without overlap. The proof employs the recent concepts of angle-monotone and radially monotone curves. The proof is constructive, leading to a polynomial-time algorithm for finding the edge-cuts, at worst O(n^2); a version has been implemented.
Cite
@article{arxiv.1707.01006,
title = {Edge-Unfolding Nearly Flat Convex Caps},
author = {Joseph O'Rourke},
journal= {arXiv preprint arXiv:1707.01006},
year = {2021}
}
Comments
34 pages, 28 figures, 16 references. Version 2 shortened the proof of Lemma 3 and added fewest nets to the Discussion section. Version 3 was a significant revision, the full version of a conference paper, reflecting corrections and suggestions from four referees. Version 4 clarifies the proof of Theorem 9 on p.11, and adds Figure 21 (p.26) to better illustrate that proof