Sibson's formula for higher order Voronoi diagrams

TL;DR

This work generalizes Sibson's local coordinate framework from the classical order-1 Voronoi diagram to higher-order diagrams $V_k(S)$. It provides a coordinate expression for a site $Q_\ell$ as a convex combination of neighbours using volume-ratio weights computed from intersections of $V_{k-1}(S)$, $V_k(S)$, and $V_{k+1}(S)$ within a bounded region $R_k(\ell)$, with $k=1$ recovering the original Sibson formula. The paper also extends these ideas to higher-order natural neighbour interpolation, deriving weighted interpolants for function values $G(Q_\ell)$ that remain convex combinations and suggesting practical benefits from using multiple orders, including improved robustness. Overall, the results offer a unified framework for higher-order coordinate representations and interpolation in Voronoi-based schemes, with potential applications in surface reconstruction and data interpolation.

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.

Figures (10)

• Figure 1: For a set $S=\{Q_1,\cdots,Q_5\}$ of five points in $\mathbb{R}^2$. $V_1(S)$ is shown in black, $V_2(S)$ in green, and $V_3(S)$ in orange colour. Each cell of $V_2(S)$ ($V_3(S)$) is labeled by the indices of its two (three) nearest points of $S$.
• Figure 2: $R_1(1)$ is the cell $f(\{Q_1\})$ in $V_1(S)$. $R_2(1)$ is the union of cells of $V_2(S)$ that have $Q_1$ as one of its two nearest neighbours. $R_1(1)\subset R_2(1)$.
• Figure 3: In $\mathbb{R}^2$. (a) The initial Voronoi diagram $V_1(S\setminus\{Q_\ell\})$ without query point $Q_\ell$. (b) Coloured areas given by the intersections of $f(\{Q_\ell\})$ and the cells of $V_1(S\setminus\{Q_\ell\})$, are the same as the ones given by the intersections of the cells of $V_2(S)$ (shown in dashed) with the cell $f(\{Q_\ell\})$.
• Figure 4: Illustrating Theorem \ref{['thm:Sibsonkface']} for $f(\{Q_2,Q_4,Q_5\})$ in $V_{3}(S)$, where $S$ is a set of six points in $\mathbb{R}^2$. In this case the equation reduces to $\sigma(A_5) Q_5 + \sigma(A_2)Q_2 = \sigma(B_1)Q_1 + \sigma(B_3)Q_3 + \sigma(B_6)Q_6$. (a) Regions $A_i$ are the cells of $V_2(S) \cap f(\{Q_2,Q_4,Q_5\})$, whose points have $Q_i$ as the third nearest neighbour of $S$. (b) Regions $B_i$ are the cells of $V_4(S) \cap f(\{Q_2,Q_4,Q_5\})$, whose points have $Q_i$ as the fourth nearest neighbour of $S$.
• Figure 5: The quadrilateral cell $f(P_k)=\square(C_{123}C_{124}C_{134}C_{234})$ of $V_k(S)$ is obtained by perpendicular bisector construction from $\{Q_1, Q_2, Q_3, Q_4\} \subset S.$ Point $H$ given by Equation (\ref{['eqn:H']}) is the intersection point of diagonals $Q_1Q_3$ and $Q_2Q_4.$ Triangles with same colour have proportional area.

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Corollary 3
• proof
• Theorem 6
• proof
• Lemma 7
• proof
• proof
• Remark