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arXiv · 2404.17422

Sibson's formula for higher order Voronoi diagrams

Mercè Claverol·Andrea de las Heras-Parrilla·Clemens Huemer·Dolores Lara
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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.

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Mercè Claverol, Andrea de las Heras-Parrilla, Clemens Huemer, Dolores Lara. 2024-04-26. Sibson's formula for higher order Voronoi diagrams. https://arxiv.org/abs/2404.17422

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