Spherical Voronoi Representation

• Spherical Voronoi representation is a method that partitions the sphere into regions based on geodesic proximity among a finite set of generators.
• It generalizes planar Voronoi diagrams to spherical geometry, employing inversion techniques for efficient computation and precise combinatorial characterization.
• Applications include modeling directional data, achieving optimal area allocation in centroidal tessellations, and enhancing clustering and visualization in non-Euclidean spaces.

A spherical Voronoi representation decomposes the surface of the sphere into regions based on proximity or optimality conditions with respect to a finite set of generators (“sites”). This construct generalizes planar Voronoi diagrams to the geometry of the $2$-sphere, leveraging geodesic distance as the metric and providing foundational structure for combinatorics, directional data modeling, visualization, and non-Euclidean clustering.

1. Mathematical Definitions and Fundamental Properties

Let $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$ denote the sphere of radius $r > 0$. For a point set $U \subset S$ of $n$ elements in general position (no three colinear on a great circle, no four cocircular), the classical (order-1) spherical Voronoi diagram partitions $S$ into $n$ faces, each defined by

$$V_i = \{ x \in S : d(x,s_i) \leq d(x,s_j) \;\; \forall j \neq i \}$$

where $d(\cdot,\cdot)$ is the geodesic (angular) distance: $d(x,y) = \arccos(x \cdot y / r^2)$.

The notion extends to order-$S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$0 spherical Voronoi diagrams. For $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$1 and each $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$2-subset $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$3, the order-$S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$4 face is

$S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$5

The complete order-$S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$6 diagram $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$7 is the union of the boundaries of these faces. The region of site $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$8 at order $S = \{ x \in \mathbb{R}^3 : \|x\| = r \}$9 is

$r > 0$0

Weighted and centroidal generalizations replace the proximity criteria with additive site weights and optimal centroidal assignment conditions (0912.3974, Śmieja et al., 2017).

2. Geometric Construction and Reduction via Inversion

An important geometric method relates spherical and planar Voronoi diagrams through 3D inversion. Fixing a pole $r > 0$1, inversion maps $r > 0$2 to a plane $r > 0$3 via

$r > 0$4

Under this transformation, sites $r > 0$5 are mapped to $r > 0$6 in the plane. Circles on $r > 0$7 not passing through $r > 0$8 invert to circles; those through $r > 0$9 become lines. The upshot is a decomposition

$U \subset S$0

where $U \subset S$1 and $U \subset S$2 are the planar order-$U \subset S$3 and order-$U \subset S$4 diagrams, respectively, glued along unbounded edges associated to the same bisectors. This enables construction of $U \subset S$5 in $U \subset S$6 time, where $U \subset S$7 is the planar diagram construction cost (Claverol et al., 2022).

3. Combinatorial Structure and Analytical Results

Order-$U \subset S$8 spherical Voronoi diagrams admit precise combinatorial characterization. With $U \subset S$9, $n$0, $n$1 denoting the numbers of vertices, edges, and faces: $n$2 Euler’s relation $n$3 holds. Vertices are split into type I ($n$4) and type II ($n$5) (Claverol et al., 2022).

In centroidal settings, a site is iteratively relocated to the centroid of its Voronoi cell using,

$n$6

where $n$7 is the area element. The Weighted Spherical Centroidal Voronoi Tessellation (WSCVT) algorithm updates both site positions and weights to minimize area error against prescribed targets, achieving exact area allocation within arbitrary tolerance $n$8 (0912.3974).

4. Spherical Voronoi as a Differentiable Partition for Directional Data

Spherical Voronoi diagrams serve as explicit, parameter-efficient, differentiable partitions of $n$9 for modeling functions defined on the sphere, such as view-dependent appearance in 3D graphics. Using $S$0 learnable sites $S$1 with per-site parameters (including temperature $S$2 and coefficient $S$3), the soft region weights are computed as

$S$4

The spherical function is then

$S$5

This framework allows gradients to flow for site, temperature, and coefficient optimization, interpolating between uniform and hard-partitioning as $S$6 varies. Region boundaries are smooth and adapt to local function complexity, avoiding artifacts such as SH ringing and capturing high-frequency content robustly (Sario et al., 16 Dec 2025).

5. Variants: Weighted, Centroidal, and Generalized Spherical Voronoi Tessellations

Weighted spherical Voronoi diagrams alter the assignment criterion to

$S$7

allowing regions of controlled area or density (0912.3974).

Centroidal variants (WSCVT) demand each site lie at its cell’s centroid, iteratively adjusting positions and weights to match area allocations. This achieves lower variance in region sizes and is essential for high-quality spherical data visualization and partitioning in non-uniform distributions (0912.3974).

Generalized Voronoi diagrams, as in the context of Spherical Wards clustering, use arbitrary symmetric dissimilarity measures and optimize partitions for within-cluster scatter adjusted by cardinalities. Assignments are based on cost differentials for cluster membership, and the resulting regions form a Voronoi-like diagram under non-Euclidean metrics (Śmieja et al., 2017).

6. Algorithmic Realizations and Computational Aspects

Classic spherical Voronoi diagrams can be constructed via inversion and planar diagram computation. For weighted/centroidal settings:

• Build the weighted Voronoi diagram (by geodesic distances minus weights)
• Compute actual cell area fractions $S$8
• Move each site to the centroid of its current cell, update weights to reduce area error
• Iterate until convergence within tolerance $S$9

For differentiable partitions (as in SV for 3DGS), forward and backward passes involve dot product calculation, temperature scaling, softmax evaluation, and parameter updates via chain rule with normalization constraints for sites (Sario et al., 16 Dec 2025).

Performance of such algorithms depends on the number of sites, mesh resolution for area approximation, and softness parameter $n$0. For clustering-based Voronoi diagrams, Hartigan-style iterative reassignment with efficient updates of within-cluster scatter is employed, followed by pruning of insufficiently populated clusters (Śmieja et al., 2017).

7. Applications and Impact

Spherical Voronoi representations are foundational in:

• Computational geometry for the analysis of distance-driven partitions on $n$1, higher-order combinatorics, and geometric optimization (Claverol et al., 2022)
• Appearance modeling for explicit 3D representations, providing a differentiable, interpretable, and highly expressive partition for view-dependent effects, outperforming global (SH) and local (SG) bases in task accuracy and efficiency (Sario et al., 16 Dec 2025)
• Data visualization: WSCVT enables spatially balanced spherical layouts for hierarchical data with exact control over region sizes, outperforming uniform subdivision and minimizing area waste (0912.3974)
• Spherical clustering, integrating geometric, probabilistic, and variational approaches for robust, scale-invariant partitioning under non-Euclidean metrics (Śmieja et al., 2017)

By combining analytic geometric structure with optimization and differentiable programming, spherical Voronoi models extend far beyond pure geometry into data-driven clustering, graphics, and information visualization.