Cartan 带状化与拓扑检测
Cartan ribbonization and a topological inspection
AI总结：
本文提出 Cartan 带概念及滚动方法，将曲面近似为内在平坦的带，通过 Gauss-Bonnet 定理简化拓扑检测，并给出环面和椭球面的带状化示例。
摘要：
我们发展了 Cartan 带的概念，并基于滚动的方法，将空间中的任意给定曲面通过内在平坦的带进行带状化和近似。滚动过程要求曲面上接触曲线的测地曲率与相应的 Cartan 发展曲线的测地曲率一致。本质上，这源于滚动过程中两个共动 Darboux 标架的定向对齐。利用闭合的接触中心曲线，我们获得闭合的近似 Cartan 带，其对带状化的总曲率积分贡献为零。这为特别简单的拓扑检测铺平了道路——它被简化为带如何相对于彼此组织其边缘的问题。Gauss-Bonnet 定理引导了对顶点的这种拓扑检测。最后，我们展示了两个曲面带状化的例子，即使用两条带对环面进行带状化，以及使用闭合曲率线作为带的中心曲线对椭球面进行带状化。
英文摘要：
We develop the concept of Cartan ribbons together with a rolling-based method to ribbonize and approximate any given surface in space by intrinsically flat ribbons. The rolling requires that the geodesic curvature along the contact curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve. Essentially, this follows from the orientational alignment of the two co-moving Darboux frames during rolling. Using closed contact center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particularly simple topological inspection -- it is reduced to the question of how the ribbons organise their edges relative to each other. The Gauss--Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using closed curvature lines as center curves for the ribbons.
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。
