Document Type
Conference Proceeding
Publication Date
6-1-2018
Publication Title
Leibniz International Proceedings in Informatics, LIPIcs
Abstract
The main result of this paper is a proof that a nearly flat, acutely triangulated convex cap C in ℝ3 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 φ < Φ with the z-axis orthogonal to the halfspace bounding plane. The size of Φ depends on the acuteness gap α: if every triangle angle is at most π/2 - α, then Φ ≈ 0.36√α suffices; e.g., for α = 3°, Φ ≈ 5°. 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); a version has been implemented.
Keywords
Polyhedra, Unfolding
Volume
99
First Page
641
Last Page
6414
DOI
10.4230/LIPIcs.SoCG.2018.64
ISSN
18688969
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
© Joseph O’Rourke
Recommended Citation
O'Rourke, Joseph, "Edge-Unfolding Nearly Flat Convex Caps" (2018). Computer Science: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/csc_facpubs/176
Comments
Archived as published.