Abstract
We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.
Abstract (translated)
我们分别从线积分和面积分的角度给出了平面曲线的圆面积不变量和曲面的球面体积不变量的计算方法。我们利用散度定理分别将面积积分和体积积分表示为线积分和面积分，并将结果推广到更高维的超曲面。所得的曲面积分可在三角网格上进行解析计算。这给出了一个简单的计算算法，用于计算不涉及离散环境空间的三角形曲面的球面体积不变量。我们讨论了在人类学中感兴趣的骨折碎片特征检测的潜在应用。
URL
https://arxiv.org/abs/1905.02176
