分段线性超曲面凸性检验的一种有效局部方法
An Efficient Local Approach to Convexity Testing of Piecewise-Linear Hypersurfaces
摘要：
我们证明了一个浸入$R^n$（$n\ge 3$）中的闭分段线性超曲面是凸体的边界当且仅当每个$(n-3)$ -面内部的每个点都有一个位于某个凸体边界上的邻域;不需要关于超曲面拓扑的任何假设。我们将Van Heijenoort（1952）关于$R^n$中的局部凸超曲面的定理推广到球面空间，从而得到了这个准则.基于该准则，给出了一个易于实现的凸性检验算法。对于$R^3$，算法所使用的算术运算的数量在顶点的数量上至多是线性的，而通常它在$(n-2)$ -面和$(n-3)$ -面之间的关联的数量上至多是线性的。当维数$n$不固定且只允许环运算时，该算法仍保持多项式。我们的方法比Mehlhorn等人（1996）和Devillers等人（1998）开发的凸性验证算法在更一般的情况下工作-例如，我们的方法不要求输入表面是可定向的，也不要求输入数据包括“以相干方式”定向的小平面的法向量。对于$R^3$，我们的算法的复杂性与以前的算法相同;对于更高的维度，似乎没有明确的赢家，但我们的方法是唯一一个容易处理的输入，其中的面法线是不知道是一致的方向或根本没有给出。此外，我们的方法可以推广到分段多项式曲面的小次数。
英文摘要：
We show that a closed piecewise-linear hypersurface immersed in $R^n$ ($n\ge 3$) is the boundary of a convex body if and only if every point in the interior of each $(n-3)$-face has a neighborhood that lies on the boundary of some convex body; no assumptions about the hypersurface's topology are needed. We derive this criterion from our generalization of Van Heijenoort's (1952) theorem on locally convex hypersurfaces in $R^n$ to spherical spaces. We also give an easy-to-implement convexity testing algorithm, which is based on our criterion. For $R^3$ the number of arithmetic operations used by the algorithm is at most linear in the number of vertices, while in general it is at most linear in the number of incidences between the $(n-2)$-faces and $(n-3)$-faces. When the dimension $n$ is not fixed and only ring arithmetic is allowed, the algorithm still remains polynomial. Our method works in more general situations than the convexity verification algorithms developed by Mehlhorn et al. (1996) and Devillers et al. (1998) -- for example, our method does not require the input surface to be orientable, nor it requires the input data to include normal vectors to the facets that are oriented "in a coherent way". For $R^3$ the complexity of our algorithm is the same as that of previous algorithms; for higher dimensions there seems to be no clear winner, but our approach is the only one that easily handles inputs in which the facet normals are not known to be coherently oriented or are not given at all. Furthermore, our method can be extended to piecewise-polynomial surfaces of small degree.
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