Spherical Voronoi Diagrams

• Spherical Voronoi diagrams are partitions of a sphere into convex regions where each point is assigned to its nearest site using geodesic distance.
• They enable efficient mesh generation and adaptive tessellations, with applications ranging from geophysical modeling to advanced graphics and neural radiance field methods.
• Recent advancements include weighted, centroidal, and differentiable variants that improve optimization stability and reconstruction accuracy in high-frequency appearance modeling.

A spherical Voronoi (SV) diagram is a partition of the unit sphere into convex regions (cells), each associated with a site (generator) on the sphere, such that every point on the sphere is assigned to the region corresponding to its closest site under the geodesic distance. This construction, central in computational geometry, mesh generation, and recent radiance field methods for graphics, generalizes to flexible adaptive tessellations, weighted and centroidal variants, arbitrary order diagrams, and differentiable forms suitable for optimization in deep learning frameworks.

1. Formal Definition and Types of Spherical Voronoi Diagrams

Let $S^2 = \{ x \in \mathbb{R}^3 : \|x\|_2 = 1 \}$ denote the unit sphere. Given a collection of $K$ sites $\{c_1, \ldots, c_K\} \subset S^2$, the (hard) spherical Voronoi cell $V_i$ associated to $c_i$ is defined as: $$V_i := \{ d \in S^2 : \text{dist}(d, c_i) \leq \text{dist}(d, c_j) \ \forall j \neq i \}$$ where $\text{dist}(d, c) = \arccos(d \cdot c)$ is the geodesic (great-circle) distance. Equivalently, since $\arccos$ is monotonic, the cell may be written as $$V_i = \{ d \in S^2 : d \cdot c_i \geq d \cdot c_j \ \forall j \neq i \}$$ (Sario et al., 16 Dec 2025).

Order-$k$ spherical Voronoi diagrams $SV_k(U)$, for a set $U$ of $n$ sites, assign to each $k$-subset $P_k \subset U$ the region of the sphere where these are the $k$ nearest sites. The cell structure is defined by great-circle bisectors between sites, and combinatorial complexity formulas for the number of faces, edges, and vertices are available for arbitrary $k$ (Claverol et al., 2022).

Weighted spherical Voronoi and centroidal tessellations further assign nonnegative weights $w_i$ to each site $c_i$, modifying the effective distance via a power diagram or area proportionality constraint, resulting in a partition whose cell areas match prescribed weights (0912.3974).

2. Structural Properties and Combinatorics

SV diagrams always partition the sphere into spherically convex regions. Edges are arcs of great circles, and vertices correspond to intersection points of three or more bisectors. For the classical ($k=1$) diagram with $n$ sites in general position, the diagram is simple and each cell is connected, convex, and star-shaped with respect to its site (Claverol et al., 2022).

For arbitrary order-$k$ spherical Voronoi diagrams:

• Each face is the intersection of $k$ closed half-spheres.
• Combinatorial complexity is given by:

$$v_k = 4kn - 4k^2 - 2n, \quad e_k = 6kn - 6k^2 - 3n, \quad f_k = 2kn - 2k^2 - n + 2$$

where $n$ is the number of sites and $k$ is the order (Claverol et al., 2022).

For a typical cell in a Voronoi tessellation generated by $n+1$ independent random sites, the $f$-vector is distributed as the reverse $f$-vector of a beta$'$ polytope in $\mathbb{R}^d$, with explicit expected face-number formulas in any dimension (Kabluchko et al., 2019).

A cycle-double-cover property holds: every edge of order-$k$ SV is covered by exactly two oriented cycles, one for each involved site, and the orientation is always opposite. This yields a small ($n$-sized), orientable, cycle double cover (Claverol et al., 2022).

3. Numerical Methods and Algorithmic Constructions

The dominant construction approach for $k=1$ SV diagrams uses convex hull computation in $\mathbb{R}^3$ of the sites (possibly with weighted augmentation), extracting the dual (Delaunay triangulation) and computing cell polygons in $O(n \log n)$ per iteration (0912.3974).

Weighted spherical centroidal Voronoi tessellation (WSCVT) algorithms optimize site locations and weights to drive the actual area fraction $a_i$ of each cell towards a prescribed $d_i = w_i / \sum_j w_j$ via an alternating update: recompute the weighted diagram, move each site to the centroid of its cell (computed via triangularization and area-weighted averaging), and adjust weights multiplicatively. Iteration continues until $\max_i |a_i - d_i|$ is below a target threshold (0912.3974).

For large-scale meshes and high-quality partitions: Lloyd-preconditioned L-BFGS (quasi-Newton) methods are efficient for centroidal SV problems, dramatically accelerating convergence over classical Lloyd and standard L-BFGS steps, especially for variable-resolution meshes. This includes robust scaling via domain decomposition on parallel architectures (Yang et al., 2017).

Order-$k$ SV diagrams are efficiently constructed via inversion in $\mathbb{R}^3$, mapping the sphere to a plane, computing order-$k$ and order-$(n-k)$ planar Voronoi diagrams, gluing along their unbounded edges, and inverting back to the sphere (Claverol et al., 2022).

4. Differentiable Soft Spherical Voronoi for Machine Learning

In radiance field learning and explicit 3D appearance modeling, a differentiable “soft” SV parameterization replaces the hard partition with a smooth softmax assignment: $$w_i(d) = \frac{\exp(\tau (s_i \cdot d))}{\sum_{j=1}^K \exp(\tau (s_j \cdot d))}$$ where $\{s_i\}$ are learnable direction vectors on $S^2$ and $\tau$ is a (learnable) temperature controlling the partition's sharpness (Sario et al., 16 Dec 2025). As $\tau \to \infty$, the soft assignment approaches a hard partition; as $\tau \to 0$, the assignment approaches a uniform blending.

The appearance at direction $d$ is reconstructed by blending site-wise appearance descriptors $c_i$: $f(d) = \sum_{i=1}^K w_i(d)\,c_i$ All parameters $\{s_i, \tau_i, c_i\}$ are fully differentiable and trained end-to-end by standard deep learning optimizers (e.g., Adam), with backward gradients readily available through automatic differentiation frameworks (Sario et al., 16 Dec 2025).

In graphics, this soft SV enables appearance representations that adaptively capture sharp, high-frequency view- and reflection-dependent effects, outperforming both bandlimited bases (spherical harmonics) and rigidly parameterized lobes (spherical Gaussians) in both empirical accuracy and optimization stability (Sario et al., 16 Dec 2025).

5. Applications: Mesh Generation and Modern 3D Appearance Modeling

Mesh Generation and Simulation

Spherical Voronoi and centroidal tessellations underlie mesh generation for geophysical, climate, and atmospheric models, supporting ultra-high resolution and adaptive, smoothly varying mesh densities. Weighted and variable-resolution centroidal tessellations achieve area-proportional parcellations and smooth gradations. Modern implementations use parallel, domain-decomposed, preconditioned quasi-Newton solvers for tractable million-point problems (Yang et al., 2017).

The WSCVT approach is broadly superior to naively subdivided “TriSphere” meshes, which can exhibit large fractions of wasted surface area (up to $70\%$ for non-power-of-four site counts), and cannot enforce general area weights (0912.3974). WSCVT achieves zero wasted area and fine control over cell areas for arbitrary numbers of sites.

3D Appearance and View Synthesis

Spherical Voronoi parameterizations provide state-of-the-art explicit appearance models in neural radiance fields and 3D Gaussian Splatting pipelines. The ability to represent both diffuse and sharp reflective effects, robustly and efficiently, derives from the SV's adaptive, non-overlapping support and direct association of directional appearance with learnable regions of the sphere (Sario et al., 16 Dec 2025).

Empirically, SV-based models achieve superior metrics (PSNR, SSIM, LPIPS) for radiance and reflection compared to spherical harmonics and spherical Gaussian models, with improved optimization stability and convergence to sharp features (Sario et al., 16 Dec 2025).

6. Random Spherical Tessellations and Typical Cell Geometry

The geometry and combinatorics of SV diagrams generated by random site processes are linked to random convex polytopes in $\mathbb{R}^d$. Kabluchko and Thäle characterized the $f$-vector distribution of a typical cell in an SV tessellation as matching that of the dual (beta$'$) polytope, allowing explicit computation of expected face numbers and further geometric observables via integral formulas (Kabluchko et al., 2019).

For dimension $d=2$, the expected number of edges (and hence vertices) in the typical cell for $n+1$ uniform random generators is $6(1 - 2/(n+1))$; for higher dimensions, more intricate integral formulae exist, but the basic correspondence with beta$'$ polytopes persists.

A direct relation allows explaining total statistics (e.g., total number of $k$-faces in the tessellation) in terms of the typical cell values: $$\mathbb{E}[f_k(m_{n,d})] = \frac{n}{d-k+1}\,\mathbb{E}[f_k(V_{n,d})]$$ where $m_{n,d}$ is the total tessellation and $V_{n,d}$ is the typical cell (Kabluchko et al., 2019).

7. Summary Table: Spherical Appearance Representations in View Synthesis

Representation	Main Advantage	Main Limitation
Spherical Harmonics (SH)	Stable orthonormal basis, simple	Band-limited, ringing, poor sharpness
Spherical Gaussians (SG)	Localized, can model peaks	Ill-conditioned if misaligned, convergence
Spherical Voronoi (SV)	Adaptive, non-overlapping, robust	Cost $O(K)$, slightly higher eval cost vs SH

SV achieves superior empirical performance in radiance and specular reflection tasks, with $+0.3\text{--}0.5$ dB PSNR gains over SH and improved robustness in complex, high-frequency effects (Sario et al., 16 Dec 2025).

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Spherical Voronoi diagrams—across their classical, weighted, centroidal, soft/differentiable, and higher-order forms—constitute a fundamental tool for partitioning the sphere. Their applications range from mesh generation to modern differentiable graphics pipelines, with theoretical foundations firmly anchored in convex geometry, combinatorics, and optimization. Ongoing research advances both efficient computation and nuanced analysis of geometric and functional properties, particularly in high-dimensional, adaptive, and data-driven contexts (Sario et al., 16 Dec 2025, Yang et al., 2017, Claverol et al., 2022, Kabluchko et al., 2019, 0912.3974).