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

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Rights

© Joseph O’Rourke

Comments

Archived as published.

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