Power Voronoi Diagram

• Power Voronoi diagrams are weighted generalizations of classical Voronoi diagrams that compare power distances instead of Euclidean distances.
• They yield convex, affine cells by subtracting squared radii, enabling efficient geometric optimization, regular triangulations, and accurate test-time adaptations.
• They underpin advanced algorithms in computational geometry and machine learning by defining decision boundaries through feature-space partitioning.

A power Voronoi diagram, also called a power diagram or Laguerre diagram, is a weighted generalization of the classical Voronoi diagram in which Euclidean space is partitioned by comparing the power distance to weighted sites. For a site with center $c_i \in \mathbb{R}^d$ and radius $r_i \ge 0$, the power distance is

$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$

and the corresponding cell is

$$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$

If all weights vanish, the construction reduces to the ordinary Voronoi diagram. Because the common quadratic term cancels in pairwise comparisons, power diagrams retain an affine cell structure while allowing site-dependent weighting, which is why they recur in generalized Voronoi theory, regular triangulations, geometric optimization, and recent machine-learning formulations (Edwards et al., 2024).

1. Definition and weighted-site formulation

The defining object of a power diagram is the power function attached to each weighted site. In the sphere formulation, a site is $S(c_i,r_i)$ and the distance comparison is

$\pi_i(x)=\|x-c_i\|^2-r_i^2.$

In the weighted-point formulation, one writes $w_i=r_i^2$, so the same cell can be expressed through $\|x-c_i\|^2-w_i$. The cell of site $i$ is therefore the minimization region of a quadratic function with a site-specific additive offset (Edwards et al., 2024).

This formulation is exactly the one used in recent test-time adaptation work under feature-space notation. In TTVD, sites are class prototypes $\mu_k$, weights are $r_i \ge 0$0, and the power distance is written

$r_i \ge 0$1

with assignment rule

$r_i \ge 0$2

Setting $r_i \ge 0$3 for all $r_i \ge 0$4 recovers the classical Voronoi diagram, so the weighted and unweighted constructions differ only by the additive squared-radius term (Lei et al., 2024).

A common misconception is to treat all weighted Voronoi diagrams as interchangeable. The power diagram is only one weighted variant. TTVD explicitly distinguishes the Laguerre–Voronoi or power-distance construction from Apollonius diagrams and Johnson–Mehl growth diagrams, which use different weighting schemes such as linear additive offsets or growth times rather than subtracted squared radii (Lei et al., 2024).

2. Affine geometry, convexity, and dual structure

Although the power distance is quadratic, the comparison between two sites is affine. Expanding

$r_i \ge 0$5

gives

$r_i \ge 0$6

Each pairwise dominance test is therefore a linear inequality, and every power cell is a convex polyhedron in $r_i \ge 0$7 (Edwards et al., 2024).

The same fact can be written as an affine decomposition

$r_i \ge 0$8

Since the common term $r_i \ge 0$9 cancels, the power diagram is equivalently the minimization diagram of the affine functions $\pi_i(x) = \|x-c_i\|^2 - r_i^2,$0. This is the basis of the classical lifting construction: one associates to each site the hyperplane

$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$1

and the vertical projection of the lower envelope of these hyperplanes yields the power diagram (Edwards et al., 2024).

In feature-space form, the same affine boundary appears in TTVD. For two classes $\pi_i(x) = \|x-c_i\|^2 - r_i^2,$2 and $\pi_i(x) = \|x-c_i\|^2 - r_i^2,$3, the boundary of the corresponding power cells is the radical hyperplane

$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$4

which expands to

$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$5

The boundaries are thus affine in the ambient feature space, and the cells remain convex under squared Euclidean distance (Lei et al., 2024).

The dual structure of a power diagram is the regular triangulation, also called the weighted Delaunay triangulation. In three dimensions, two weighted sites are adjacent exactly when their cells share a polygonal face, and weighted sites may have empty cells, a phenomenon absent in the ordinary unweighted Voronoi setting (Taveira et al., 7 May 2026). This duality is classical and is recovered by the same lifting machinery used for the affine-envelope representation (Edwards et al., 2024).

3. Generalized and anisotropic variants

Power diagrams occupy a specific place within a broader hierarchy of generalized Voronoi diagrams. In the Lie sphere framework, they appear both as diagrams defined by extremal spheres in Lie sphere space and as minimization diagrams of functions that become affine after lifting to a higher-dimensional space. In that unifying description, classical Voronoi diagrams, power diagrams, order-$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$6 diagrams, farthest-point diagrams, Apollonius diagrams, medial axes, and diagrams with mixed point, sphere, and half-space sites all fall under a common polyhedral or affine reduction (Edwards et al., 2024).

A more general anisotropic extension is the generalized balanced power diagram (GBPD), defined in two dimensions by

$\pi_i(x) = \|x-c_i\|^2 - r_i^2,$7

where $\pi_i(x) = \|x-c_i\|^2 - r_i^2,$8 is symmetric positive definite and $\pi_i(x) = \|x-c_i\|^2 - r_i^2,$9. When $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$0, the model reduces exactly to the classical power diagram: $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$1 When the matrices vary by site, the bisectors are no longer generally affine. The boundary between cells $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$2 and $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$3 is the conic

$$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$4

so GBPD cells can have curved edges, need not be convex, and may even be disconnected (Jung et al., 2023).

This distinction is important: the straight-edged, convex-polyhedral structure is a defining characteristic of the classical power diagram, not of all weighted or generalized Voronoi constructions. Jung and Redenbach’s analytic treatment of the 2D GBPD makes this explicit by deriving vertex and edge representations for the anisotropic case, while also showing that the isotropic $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$5 specialization restores the usual Laguerre geometry with line bisectors and convex polygons (Jung et al., 2023).

4. Algorithms and computational models

Several algorithmic routes compute power diagrams by exploiting their affine structure. In the generalized Voronoi framework, one can either form half-space inequalities in a lifted space and compute an intersection polyhedron, or build the arrangement of affine functions obtained after lifting and extract the relevant minimization diagram. For the Lie-sphere-based algorithm, the stated complexity is

$$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$6

which yields $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$7 worst-case behavior in 2D and $$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$8 in 3D (Edwards et al., 2024).

On triangle meshes, the surface-based power diagram replaces Euclidean distance by geodesic distance. SurfaceVoronoi defines the geodesic power distance

$$\mathrm{Cell}_i = \{\,x \in \mathbb{R}^d : \pi_i(x) \le \pi_j(x)\ \forall j\,\}.$$9

for sites $S(c_i,r_i)$0 on a mesh surface $S(c_i,r_i)$1, and constructs each per-triangle diagram by interpolating squared distances at triangle vertices as planes in local coordinates. On planar triangle meshes this squared-distance linearization is exact, and subtracting the weights preserves the affine equalities, so the same construction gives the exact planar power diagram. Inside each triangle, power bisectors are straight segments, while on curved surfaces the method yields a piecewise-linear approximation driven by the same weighted lower-envelope principle (Xin et al., 2022).

Large-scale three-dimensional computation has recently been addressed with a direct GPU clipping algorithm. In that formulation, each cell is built independently from an initial convex region by clipping with weighted bisecting planes induced by candidate neighbors. The method uses a BVH augmented with per-node weight maxima, a best-first traversal strategy, and a directional culling criterion based on support bounds of the evolving cell. On real scenes with $S(c_i,r_i)$2M–$S(c_i,r_i)$3M sites, the reported runtimes for Voronoi diagrams are $S(c_i,r_i)$4–$S(c_i,r_i)$5 s on RTX 5090 and $S(c_i,r_i)$6–$S(c_i,r_i)$7 s on H200; for the weighted case they are $S(c_i,r_i)$8–$S(c_i,r_i)$9 s and $\pi_i(x)=\|x-c_i\|^2-r_i^2.$0–$\pi_i(x)=\|x-c_i\|^2-r_i^2.$1 s, respectively. The same work reports that BVH construction accounts for approximately $\pi_i(x)=\|x-c_i\|^2-r_i^2.$2 of total time and the fused kernel approximately $\pi_i(x)=\|x-c_i\|^2-r_i^2.$3, with clipping dominating about $\pi_i(x)=\|x-c_i\|^2-r_i^2.$4 of kernel time (Taveira et al., 7 May 2026).

These algorithmic developments all rely on the same invariant: once the power bisector is written as an affine half-space, the diagram can be recovered by polyhedral intersection, lower-envelope extraction, or repeated clipping. The primary differences concern domain geometry, distance oracle, and scale.

5. Role in test-time adaptation and feature-space partitioning

The 2024 paper "TTVD: Towards a Geometric Framework for Test-Time Adaptation Based on Voronoi Diagram" formalizes a class of neighbor-based test-time adaptation methods as Voronoi-type partitions in feature space and then refines them with power diagrams (Lei et al., 2024).

In TTVD, class means computed offline from training data serve as sites $\pi_i(x)=\|x-c_i\|^2-r_i^2.$5, and the power-diagram weights are extracted from a pretrained multinomial logistic classifier through the constructive mapping

$\pi_i(x)=\|x-c_i\|^2-r_i^2.$6

where $\pi_i(x)=\|x-c_i\|^2-r_i^2.$7 and $\pi_i(x)=\|x-c_i\|^2-r_i^2.$8 are the linear classifier’s weights and bias. This gives a direct way to derive the weighted Voronoi geometry of the classifier in feature space. Soft labels are then produced by a softmax over negative distances or influences, and entropy minimization updates only normalization affine parameters during test-time adaptation.

TTVD introduces two related cluster-based generalizations. The Cluster-induced Voronoi Diagram (CIVD) assigns each class a set of sites $\pi_i(x)=\|x-c_i\|^2-r_i^2.$9, obtained for example from self-supervised rotations $w_i=r_i^2$0, and aggregates them by

$w_i=r_i^2$1

The Cluster-induced Power Diagram (CIPD) inserts the per-class power weights into the same aggregation: $w_i=r_i^2$2 The paper states that $w_i=r_i^2$3, with $w_i=r_i^2$4 used in experiments, de-emphasizes far-away sites and stabilizes adaptation from small batches.

A second geometric device is noisy-sample filtering by “diagram subtraction.” TTVD constructs both a Voronoi diagram and a power diagram in feature space, subtracts PD cells from VD cells to form a boundary band, and excludes samples in that band from adaptation updates. The motivation given is that entropy-based filtering often misses noisy samples unless they sit exactly at the boundary, whereas the power diagram shifts boundaries and enlarges the high-risk region near decision surfaces.

The reported empirical gains are framed in both accuracy and calibration. Under TTAB’s standardized protocol on CIFAR-10-C, CIFAR-100-C, ImageNet-C, and ImageNet-R, TTVD reports classification error reductions of $w_i=r_i^2$5, $w_i=r_i^2$6, $w_i=r_i^2$7, and $w_i=r_i^2$8, and ECE reductions of $w_i=r_i^2$9, $\|x-c_i\|^2-w_i$0, $\|x-c_i\|^2-w_i$1, and $\|x-c_i\|^2-w_i$2, respectively. On CIFAR-10-C level 5, the ablation from VD to CIVD to CIPD yields $\|x-c_i\|^2-w_i$3, $\|x-c_i\|^2-w_i$4, and $\|x-c_i\|^2-w_i$5 error. On ImageNet-C motion blur, the reported errors are TTVD $\|x-c_i\|^2-w_i$6, AdaNPC $\|x-c_i\|^2-w_i$7, TAST $\|x-c_i\|^2-w_i$8, and T3A $\|x-c_i\|^2-w_i$9. The paper also reports robustness to prototype estimation noise on ImageNet: using $i$0, $i$1, or $i$2 of the training data to compute sites gives an average error of approximately $i$3–$i$4.

These results do not change the mathematical definition of a power diagram, but they show that the weighted partition is not merely descriptive. In this setting it becomes an operational object that shapes decision regions, determines soft labels, and defines an explicit exclusion band for unstable samples.

6. Hyperbolic reductions and clipped power constructions

Power diagrams also serve as an exact computational intermediary for certain non-Euclidean Voronoi structures. In the Klein projective ball model of hyperbolic space, the hyperbolic Voronoi diagram is affine and can be obtained by clipping a Euclidean power diagram. For a site $i$5 in the Klein ball, the corresponding Euclidean weighted object is

$i$6

The power bisectors of these mapped balls coincide with the Klein hyperbolic bisectors, and intersecting the resulting power cells with the Klein domain yields the hyperbolic Voronoi diagram. The stated combinatorial complexity is $i$7, with optimal construction time $i$8 (Nielsen et al., 2012).

The same paper identifies an arithmetic limitation of the Klein mapping: it involves square roots and thus algebraic arithmetic even for rational input. In the Beltrami hemisphere model, this difficulty is removed by working on an affine chart and mapping a hemisphere point $i$9 to a power site

$\mu_k$0

Because this construction avoids square roots, it supports rational arithmetic while preserving the reduction to a clipped power diagram (Nielsen et al., 2012).

Further results extend the same idea to $\mu_k$1-order hyperbolic diagrams. In the Klein model, a $\mu_k$2-subset $\mu_k$3 of sites is mapped to

$\mu_k$4

and the $\mu_k$5-order hyperbolic Voronoi diagram is then the clipping of the Euclidean power diagram of these weighted objects by the Klein ball. The same work interprets the construction through a concave lifting potential, the northern hemisphere $\mu_k$6, which plays the role that the paraboloid plays for Euclidean Voronoi and power diagrams (Nielsen et al., 2014).

This clipped-power viewpoint makes a broader geometric point. Power diagrams are not limited to weighted Euclidean nearest-neighbor partitioning; they also function as an affine computational substrate for generalized, order-$\mu_k$7, and hyperbolic Voronoi constructions once the appropriate lifting or projective model has been chosen.