Sibson's formula for higher order Voronoi diagrams

TL;DR

This paper generalizes Sibson's formula for higher order Voronoi diagrams, enabling the expression of points as convex combinations of other points using ratios of volumes from Voronoi diagrams of any order.

Contribution

The paper extends Sibson's original formula to higher order Voronoi diagrams, broadening the applicability of natural neighbor interpolation methods.

Findings

Generalized Sibson's formula for any order Voronoi diagram.

Provides a new method for expressing points as convex combinations.

Enhances interpolation techniques using higher order Voronoi structures.

Abstract

Let $S$ be a set of $n$ points in general position in $\mathbb{R}^d$. The order-$k$ Voronoi diagram of $S$, $V_k(S)$, is a subdivision of $\mathbb{R}^d$ into cells whose points have the same $k$ nearest points of $S$. Sibson, in his seminal paper from 1980 (A vector identity for the Dirichlet tessellation), gives a formula to express a point $Q$ of $S$ as a convex combination of other points of $S$ by using ratios of volumes of the intersection of cells of $V_2(S)$ and the cell of $Q$ in $V_1(S)$. The natural neighbour interpolation method is based on Sibson's formula. We generalize his result to express $Q$ as a convex combination of other points of $S$ by using ratios of volumes from Voronoi diagrams of any given order.