Spherical Voronoi Diagrams

• Spherical Voronoi diagrams are partitions of the unit sphere where each region is defined by the nearest site using geodesic (great-circle) distance.
• They feature multiple variants—including order-k, weighted, and centroidal forms—each tailored for specific geometric properties and computational strategies.
• Construction methods such as inversion, Delaunay triangulation, and iterative Lloyd schemes enable efficient implementations for applications in geodesy, graphics, and spatial analysis.

A spherical Voronoi diagram is a partition of the unit sphere $S^2 \subset \mathbb{R}^3$ into regions, each associated with a given site (point) such that every location on the sphere is assigned to its nearest site under the geodesic (great-circle) distance. This structure generalizes planar Voronoi diagrams to spherical geometry, with modifications required by the non-Euclidean nature of the sphere. Spherical Voronoi diagrams appear in pure mathematics, computational geometry, spatial statistics, geodesy, computer graphics, and geospatial data analysis. Both unweighted and weighted versions exist, as do higher-order generalizations, centroidal configurations, and stochastic and combinatorial analyses of expected structural properties (Claverol et al., 2022, 0912.3974, Kabluchko et al., 2019).

1. Fundamental Definitions and Variants

A spherical Voronoi diagram is defined with respect to $n$ distinct sites $U = \{u_1, \dots, u_n\} \subset S^2$. The geodesic distance on the unit sphere is $d_S(x, y) = \arccos(\langle x, y \rangle)$ for $x, y \in S^2$.

• Order-1 (Nearest-neighbor) Spherical Voronoi Diagram: The cell associated with $u_i$ is $$V(u_i) = \{ x \in S^2 : d_S(x, u_i) \leq d_S(x, u_j)\,\,\forall j \neq i \}$$. Each $V(u_i)$ is a convex spherical polygon bounded by great-circle bisectors (Claverol et al., 2022).
• Order-$k$ Spherical Voronoi Diagram: For $1 \leq k < n$, the order-$k$ diagram $SV_k(U)$ assigns to each $k$-element subset $P \subset U$ the region $f(P) = \{ x \in S^2 : \text{the set of$k$nearest sites to$x$is exactly$P$}\}$, using lexicographic ordering of distances for tie-breaking. Faces correspond to combinations of $k$ nearest neighbors (Claverol et al., 2022).
• Weighted Spherical Voronoi Diagram (WSVD): Each site $s_i$ receives a positive weight $w_i$. The (additive) weighted distance is $D_i(x) = w_i\,d_S(x, s_i)$. The weighted cell for $s_i$ is $$V_i = \{x \in S^2: w_i d_S(x, s_i) \leq w_j d_S(x, s_j)\,\,\forall j\}$$ (0912.3974).
• Centroidal Spherical Voronoi Tessellations (CVT): A configuration where each site is at the centroid of its Voronoi cell. The centroid is the area-weighted mean of the region, and in the weighted setting (WSCVT) the centroid condition integrates the weights over the cell (0912.3974).

The unweighted and weighted cases differ in how critical curves (cell boundaries) are defined, with weights introducing anisotropy.

2. Construction Methods and Algorithmic Frameworks

Construction of spherical Voronoi diagrams can proceed by direct geometric means or via reduction to planar Voronoi diagrams using inversion.

• Inversion-Based Construction: For $SV_1(U)$, choose a center of inversion $i \in S^2$ not coincident with any site and apply the inversion in $\mathbb{R}^3$ to map $S^2$ onto a plane. Great-circle bisectors become perpendicular bisectors in the plane. This process homeomorphically relates $SV_1(U)$ with the juxtaposition of nearest- and farthest-neighbor planar Voronoi diagrams, glued along their shared unbounded edges. For $SV_k(U)$, the faces outside (respectively, inside) the region of points with $i$ among their $k$ (resp. $n-k$) nearest sites correspond to faces of the planar order-$k$ (resp. order-$(n-k)$) diagram in the image plane under inversion. Thus, $SV_k(U)$ can be constructed in linear time over the planar diagrams’ complexities (Claverol et al., 2022).
• Delaunay-Based Methods: Spherical Delaunay triangulation via the convex hull in $\mathbb{R}^3$ yields dual adjacency for Voronoi cells. For the weighted case, weighted circumcenters (where $w_a d(a,x) = w_b d(b,x) = w_c d(c,x)$ for triangle $(a,b,c)$) are used to determine cell vertices. The dual graph encodes Voronoi adjacency structure (0912.3974).
• Iterative Lloyd-Type Schemes (for CVT/WSCVT): The weighted centroidal Voronoi tessellation is computed by alternately updating sites to centroids of current cells and reconstructing the Voronoi diagram until convergence (maximum area error below tolerance $\epsilon$). Weights may be adaptively adjusted to target proportional surface areas. Each iteration involves convex hull/Delaunay computation, circumcenter determination, cell construction, centroid update, and optional weight normalization (0912.3974).

A summary of computational complexities and differences:

Construction/Algorithm	Complexity	Features
Planar SV reduction (inversion)	$O(n \log n + k(n-k))$	Utilizes high-quality planar codes; order-$k$ and $(n-k)$ (Claverol et al., 2022)
Delaunay-based (S²)	$O(n \log n)$ per iteration	Weighted circumcenters, CVT support (0912.3974)
TriSphere (icosahedral grid)	$O(n)$ subdivision	Uniform only for $n=20\cdot 4^k$; inefficient for arbitrary $n$ (0912.3974)

3. Combinatorial Structure and Expected Properties

The combinatorics of spherical Voronoi diagrams, particularly for arbitrary order and for random point configurations, have been rigorously characterized.

• Vertex, Edge, and Face Counts (General Position): For a set $U$ of $n$ sites in general position (no three collinear, no four cocircular):
  • Number of Type I vertices in $SV_k(U)$: $2k(n-k-1)$
  • Number of Type II vertices in $SV_k(U)$: $2(k-1)(n-k)$
  • Total vertex count: $V_k = 4kn - 4k^2 - 2n$
  • Edge count: $E_k = (3/2)V_k = 6kn - 6k^2 - 3n$
  • Face count: $F_k = 2kn - 2k^2 - n + 2$
  • (Claverol et al., 2022)
• Structure of Vertices: Type I vertices correspond to centers of circles through three sites enclosing exactly $k-1$ other sites; Type II vertices correspond to circles through three sites enclosing exactly $k-2$ other sites (Claverol et al., 2022).
• Typical Cell Statistics: For a set of $n+1$ i.i.d. uniform points on $S^d$, the typical Voronoi cell’s $f$-vector $(f_0,\dotsc,f_{d-1})$ (number of $k$-faces for $k=0$ to $d-1$) is distributed as the reversed $f$-vector of a beta′ polytope in $\mathbb{R}^d$ with parameter $\beta = d$ (Kabluchko et al., 2019).
  • For $d=2$ and large $n$, the expected number of vertices/edges in a typical cell is $6\frac{n-1}{n+1}$, tending to $6$ as $n \to \infty$, which coincides with the planar Poisson–Voronoi value (Kabluchko et al., 2019).
  • For $d=3$, explicit integral formulae provide expected $f$-vectors (Kabluchko et al., 2019).
  • The total number of $k$-faces in the complete tessellation is $E[N_k] = \frac{n+1}{d-k+1} E[f_k(V_{n+1,d})]$ (Kabluchko et al., 2019).

4. Weighted and Centroidal Generalizations

Weighted spherical Voronoi diagrams and centroidal variants accommodate site importance or desired surface area allocations.

• Weighted Cell Definition: Each site $s_i$ with weight $w_i$ defines the cell $$V_i = \{ x \in S^2 : w_i d(x, s_i) \leq w_j d(x, s_j) \text{ for all } j \}$$ (0912.3974).
• Weighted Centroidal Constraint: In a WSCVT, each $s_i$ is updated to the (weighted) centroid of its cell, computed as

$$s_i = \left( \int_{V_i} w_i\, x\, dA(x) \right) / \left( \int_{V_i} w_i\, dA(x) \right)$$

(0912.3974).

• Area Adjustment: To enforce proportional area allocation, weights are iteratively updated based on the area error between actual and desired cell proportions. The process converges (in practice, within 20–100 iterations for typical tolerances) to a configuration matching specified area allocations (0912.3974).
• Energy Functional: The centroidal Voronoi arrangement minimizes the energy

$$E(\{s_i\}) = \sum_{i=1}^n \int_{V_i} w_i\, d(x, s_i)^2\, dA(x)$$

(0912.3974).

• Practical Implementation: Each iteration requires Delaunay triangulation, weighted circumcenter computation, cell area and centroid evaluation, and (optionally) weight renormalization. Representation uses arrays for sites/weights and face-incidence lists for triangulation (0912.3974).

5. Structural and Topological Properties

Spherical Voronoi diagrams possess several notable properties.

• Cycle Double Cover: Each edge is labeled by the two generating sites. For fixed $i$, the edges labeled $i$ form a simple cycle $B_k(i)$ which winds clockwise around the region $R_k(i)$. The $n$ cycles $\{B_k(i): i \in U\}$ provide a small orientable cycle double cover of $SV_k(U)$; each edge is traversed once in each direction in this covering (Claverol et al., 2022).
• Planarity and Regularity: $SV_k(U)$ is a 3-regular planar graph (all vertices have degree 3) in general position (Claverol et al., 2022).
• General Position and Degeneracies: Under generic assumptions (no three collinear, no four cocircular), all faces are spherical polygons and higher-order degeneracies are absent. Degenerate configurations require perturbation or case-specific logic, mirroring methods from the planar setting (Claverol et al., 2022).
• Asymptotic Behavior: As the number of sites increases, the typical spherical cell converges (under stereographic projection and rescaling) to the typical cell of a Poisson–Voronoi tessellation in $\mathbb{R}^d$ (Kabluchko et al., 2019).

6. Computational Complexity and Algorithmic Comparison

The computational cost of constructing spherical Voronoi diagrams varies by algorithm and diagram type.

Approach	Asymptotic Time	Notes
Planar order-$k$, inversion	$O(n \log n + k(n-k))$	Order-$k$ and $(n-k)$ planar diagrams, inversion overhead $O(n)$ (Claverol et al., 2022)
Delaunay-based, centroidal	$O(n \log n)$ per iteration	Lloyd-type iteration for (W)CVT (0912.3974)
TriSphere	$O(n)$ subdivision	Coarse for arbitrary $n$; only exact when $n=20 \cdot 4^k$ (0912.3974)

In centroidal settings, 20–100 iterations are typically required for convergence to modest area error tolerances (0912.3974).

7. Applications and Research Context

Spherical Voronoi diagrams are instrumental in diverse scientific and engineering domains:

• Geodesy and Cartography: Partitioning Earth's surface by proximity to meteorological stations, towers, or sensors (Claverol et al., 2022).
• Computer Graphics: Distributing sample points on $S^2$, blue-noise sampling for environment maps and particle systems (Claverol et al., 2022).
• Geospatial Analysis: Nearest-neighbor queries, network coverage planning, planet-scale facility location (Claverol et al., 2022).
• Materials Science: Analyzing Brillouin zones, crystal lattice structures on the sphere (Claverol et al., 2022).
• 3D Visualization: Spherical layout of tree structures and feature-rich graphs, with area-proportional allocations via WSCVT for improved data visual metaphors (0912.3974).

The reduction of spherical to planar Voronoi diagram construction enables leveraging established planar algorithms, unifying code bases for both nearest/farthest neighbor partitions and facilitating efficient implementation (Claverol et al., 2022).

A major result is the explicit identification of the typical cell's $f$-vector distribution with that of beta′ polytopes, which, in combination with integral formulae for expected face counts, provides a probabilistic geometric foundation for the asymptotic structure of spherical tessellations (Kabluchko et al., 2019).