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Edge-Unfolding Nearly Flat Convex Caps O'Rourke, Joseph polyhedra unfolding 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Joseph O'Rourke 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.64 urn:nbn:de:0030-drops-87777 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.64 eng https://creativecommons.org/licenses/by/3.0/legalcode