边可展开的近平凸帽
Edge-Unfolding Nearly Flat Convex Caps
摘要：
本文的主要结果是证明了R^3中一个近似平坦的、锐角三角形的凸帽C在平面上有一个边展开为一个不重叠的多边形。凸帽是凸多面体的曲面与半空间的交线。“几乎平坦”意味着每个外表面范式与正交于半空间边界平面的z轴形成足够小的角度Φ。Φ的大小取决于锐度间隙α：如果每个三角形角至多为π/2-α，则Φ = 0.36 sqrt（α）就足够了;例如，当α = 3 °时，φ = 5 °。即使C通过在C下添加凸多边形底而接近于多面体，该多面体也可以边展开而不重叠。证明采用最近的概念，角度单调和径向单调曲线。证明是建设性的，导致一个多项式时间算法找到边切割，在最坏的O（n^2）;一个版本已经实现。
英文摘要：
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.
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