发表机构
TU Munich（慕尼黑工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对平面各向异性 Voronoi 图，推导了星凸单元的任意阶 Minkowski 张量半解析公式，利用接触函数和微分同胚变换实现几何解析处理，并支持数值近似。
AI 中文摘要
各向异性 Voronoi 图，其中每个单元是从其站点开始生长的椭圆最先到达的区域，会产生非凸单元，其积分几何可以使用任意阶的 Minkowski 张量来描述。在这项工作中，我们推导了在成核点处星凸的平面各向异性 Voronoi 单元的 0 阶、1 阶和 2 阶任意阶 Minkowski 张量的半解析公式。所得公式涉及接触函数的积分，该函数通过微分同胚变换描述单元边界。这种变换使得几何便于解析处理，使我们能够获得所需 Minkowski 张量的表达式。尽管大多数积分缺乏闭式解，但一旦接触函数及其导数沿目标单元的边界已知，就可以通过数值方法高效近似。
英文摘要
Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.
Comments28 pages, 9 figures
