Edge-Unfolding Nearly Flat Convex Caps

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 ẑ-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◦. 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(n2); a version has been implemented. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems. G.2.2 Graph Theory.