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.
翻译:我们提出一个新的以巴中心为中心进行内插的理论框架,使用最近在数学物理学中形成的概念。通用的巴中心坐标与谢帕德的方法定义相似,使用正几何相近,即具有与其边界自然相关的合理功能的子集。正几何对准,将implices和 convex多面形的某些特性概括为各种几何天体。我们的框架统一了以前的若干构造,包括从连接和双多面体的角度界定多面体上的Wachspress坐标。我们还讨论了3D线空间、平均值坐标和样条内插的潜在应用。