We propose a novel theoretical framework for barycentric interpolation, using concepts recently developed in mathematical physics. Generalized barycentric coordinates are defined similarly to Shepard's method, using positive geometries - subsets which possess a rational function naturally associated to their boundaries. Positive geometries generalize certain properties of simplices and convex polytopes to a large variety of geometric objects. Our framework unifies several previous constructions, including the definition of Wachspress coordinates over polytopes in terms of adjoints and dual polytopes. We also discuss potential applications to interpolation in 3D line space, mean-value coordinates and splines.

中文翻译：

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重心插值的正几何

我们使用最近在数学物理学中发展起来的概念，为重心插值提出了一种新颖的理论框架。广义重心坐标的定义类似于Shepard的方法，使用的是正几何形状-具有与边界自然相关的有理函数的子集。正几何形状将单纯形和凸多面体的某些属性推广到各种各样的几何对象。我们的框架统一了以前的几种构造，包括Wachspress坐标在邻接点和双重多边形上的定义。我们还将讨论在3D线空间，平均值坐标和样条曲线中插值的潜在应用。