任意表面的曲率用于离散引力和对于d=2纯单纯复形
Curvature of an Arbitrary Surface for Discrete Gravity and for $d=2$ Pure Simplicial Complexes
AI总结：
本文提出了一种计算任意二维表面曲率的方法，用于离散引力理论，并通过高效算法计算纯单纯复形中的曲率。
摘要：
我们提出了一种计算任意三维物体二维表面曲率的方法，这为离散引力理论提供了贡献，并可能应用于网络几何。我们首先将每个待研究表面的点与其四个最近邻点联系起来，形成四边形。然后我们关注d=2的单纯形，即嵌入这些四边形中的三角形，这些构成了一个d=2的纯单纯复形。这使我们能够数值计算局部度量以及二面体，进而推导出在每个三角形或面上定义的离散曲率。我们提供了一个具有O(N log N)复杂度的高效算法，该算法首先对二维表面进行定向，解决由扭力条件产生的非线性方程组，并返回每个面上的曲率值。
英文摘要：
We propose a computation of curvature of arbitrary two-dimensional surfaces of three-dimensional objects, which is a contribution to discrete gravity with potential applications in network geometry. We begin by linking each point of the surface in question to its four closest neighbors, forming quads. We then focus on the simplices of $d=2$, or triangles embedded in these quads, which make up a pure simplicial complex with $d=2$. This allows us to numerically compute the local metric along with zweibeins, which subsequently leads to a derivation of discrete curvature defined at every triangle or face. We provide an efficient algorithm with $\mathcal{O}(N \log{N})$ complexity that first orients two-dimensional surfaces, solves the nonlinear system of equations of the spin-connections resulting from the torsion condition, and returns the value of curvature at each face.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
