论文标题
多面体的第一矩被抛物线剪切
First moments of a polyhedron clipped by a paraboloid
论文作者
论文摘要
我们提供了由抛物线片剪切的多面体的第一矩（即体积和体积加权质心）的封闭形式表达式，即抛物面的多面体与位于抛物面的一侧的三维真实空间的子集相交。这些封闭形式的表达式是根据发散定理的连续应用以及多面体面部与抛物面的相交的合理参数化的连续应用。我们提供了识别模棱两可的离散交叉拓扑的手段，并提出了防止其发生的纠正措施。最后，我们将提议的封闭形式表达式和数值方法放在测试中，以数百万随机和手动工程的多面体/抛物线相交配置。这些测试的结果表明，我们能够以计算成本以一个数量级的计算成本提供强大的机器准确估计值，该计算成本在最新的半空间剪辑算法的一个数量级范围内。
We provide closed-form expressions for the first moments (i.e., the volume and volume-weighted centroid) of a polyhedron clipped by a paraboloid, that is, of a polyhedron intersected with the subset of the three-dimensional real space located on one side of a paraboloid. These closed-form expressions are derived following successive applications of the divergence theorem and the judicious parametrization of the intersection of the polyhedron's faces with the paraboloid. We provide means for identifying ambiguous discrete intersection topologies, and propose a corrective procedure for preventing their occurence. Finally, we put our proposed closed-form expressions and numerical approach to the test with millions of random and manually engineered polyhedron/paraboloid intersection configurations. The results of these tests show that we are able to provide robust machine-accurate estimates of the first moments at a computational cost that is within one order of magnitude of that of state-of-the-art half-space clipping algorithms.
