  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 phi < Phi with the z-axis orthogonal to
the halfspace bounding plane. The size of Phi depends on the acuteness gap
alpha: if every triangle angle is at most pi/2-alpha, then Phi ~= 0.36
sqrt(alpha) suffices; e.g., for alpha ~= 3deg, Phi = 5deg. 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(n^2); a
version has been implemented.
