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Edge-Unfolding Nearly Flat Convex Caps

Author Joseph O'Rourke

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  • polyhedra
  • unfolding

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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 f < F with the z^-axis orthogonal to the halfspace bounding plane. The size of F depends on the acuteness gap a: if every triangle angle is at most pi/2 {-} a, then F ~~ 0.36 sqrt{a} suffices; e.g., for a=3°, F ~~ 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^2); a version has been implemented.

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Joseph O'Rourke. Edge-Unfolding Nearly Flat Convex Caps. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 64:1-64:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.64

Author Details

Joseph O'Rourke

References

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