Generalized Laguerre Cells

• Generalized Laguerre cells are geometric tessellations that extend classical power diagrams by incorporating weighted restrictions and intersecting with spherical boundaries.
• They are constructed analytically through power diagram clipping and pyramid decomposition, enabling precise volume and area computations in complex simulations.
• These cells play a vital role in optimal transport, fluid simulation, and anisotropic material modeling, and also find applications in algebraic representations in Lie theory.

Generalized Laguerre cells, also referred to as power cells or generalized weighted Voronoi cells, are geometric constructs arising in computational geometry, optimal transport, and the mathematical modeling of materials and fluids. They generalize classical Laguerre (power) diagrams by introducing further restrictions—such as intersection with a sphere—or by generalizing the underlying metric or shape (e.g., to ellipsoidal metrics), resulting in a much broader class of tessellations. These structures have found key applications in simulation, analysis of cellular materials, and representation of indecomposable Lie algebra modules.

1. Mathematical Formulation of Generalized Laguerre Cells

Let $\{x_i \in \mathbb{R}^d\}_{i=1}^n$ denote a set of $n$ sites with associated non-negative weights $\psi_i$. The power distance from a point $x \in \mathbb{R}^d$ to a site $(x_i, \psi_i)$ is defined by

$\pi_i(x) = \|x - x_i\|^2 - \psi_i.$

The classical (unrestricted) Laguerre cell is

$$V_i = \{ x \in \mathbb{R}^d \mid \pi_i(x) \le \pi_j(x),\, \forall j \}.$$

The generalized Laguerre cell is further obtained by restricting this polyhedral cell to the intersection with a ball of radius $\sqrt{\psi_i}$,

$$V_i^g = V_i \cap B(x_i, \sqrt{\psi_i}) = \{ x \in V_i \mid \|x - x_i\|^2 \leq \psi_i \}.$$

This restriction is crucial for partial optimal transport applications where each cell must contain a prescribed measure (e.g., volume in fluid simulation) (Plateau--Holleville et al., 9 Jan 2026).

In broader formulations, generalized Laguerre-type cells may arise from more general metrics. In $\mathbb{R}^2$, given generators $(c_i, A_i, w_i)$ where $A_i$ are positive-definite $2\times 2$ matrices, one defines the generalized (balanced) power cell as

$$C_i = \{ x \in \mathbb{R}^2 \mid d_i(x) \leq d_j(x) \,\, \forall j \neq i \}, \quad d_i(x) = (x - c_i)^T A_i (x - c_i) + w_i,$$

so that the cells may be non-convex with curved (conic) boundaries (Jung et al., 2023).

2. Analytic Geometry and Volume/Area Computation

The intersection of a classical Laguerre cell with a ball leads to nontrivial cell geometry, particularly for the computation of volumes and facet areas. Analytic decomposition is achieved by subdividing $V_i^g$ into "pyramids" $P_{ij}$, each with apex $x_i$ and base either a clipped planar facet or a spherical patch. The divergence theorem, leveraging the vector field $F(x) = (x-x_i)/3$, yields the volume of each pyramid as $h_{ij} \, |B_{ij}|$, where $h_{ij}$ is the signed distance from $x_i$ to the base and $|B_{ij}|$ the area of the base.

For planar facets: $$h_{ij} = \frac{\|x_j - x_i\|^2 + \psi_i - \psi_j}{2 \|x_j - x_i\|},$$ while for spherical bases, $h_{ij} = \sqrt{\psi_i}$. The total volume is

$$|V_i^g| = \sum_{j \in \mathcal{N}_i} h_{ij} |B_{ij}| + \sqrt{\psi_i} |\Sigma_i \cap V_i^g|,$$

where the spherical patch area on $\Sigma_i = \{ \|x - x_i\| = \sqrt{\psi_i} \}$ is

$$|\Sigma_i \cap V_i^g| = 4\pi \psi_i - \sum_{j \in \mathcal{N}_i} |\bar K_{ij}|.$$

Facet areas are computed by triangulating clipped polygons and circular sectors; spherical patches from polyhedral occlusion (Plateau--Holleville et al., 9 Jan 2026).

In generalized balanced power diagrams, edge and vertex geometry is analytic: bisectors between cells are conics, with explicit parametric representation via diagonalization in homogeneous coordinates. Intersections of conics define vertices, leading to rigorous constructions of curved cell boundaries in $\mathbb{R}^2$ (Jung et al., 2023).

3. Algorithmic Construction

For the analytic construction in three dimensions, the algorithm is as follows:

• Build the power diagram to compute neighbors and obtain planar facets.
• For each facet, clip the polygon to the sphere around the center.
• Compute planar area and project the facet onto the spherical surface to obtain the "occluded" area.
• Sum the pyramid volumes and spherical patch volume per cell.
• No convex-hull or global clipping library is required; only 2D polygon/sphere intersection and polygon-area procedures are needed.

The complexity per cell is $O(k)$ per facet, where $k$ is the number of facet vertices (typically small), and the global power diagram may be constructed in $O(n \log n)$ time using GPU-optimized routines. This approach is $2\times$–$3\times$ faster than traditional convex-clipping and admits precise, meshless representation (Plateau--Holleville et al., 9 Jan 2026).

In two-dimensional GBPDs, the algorithm evaluates all triple intersections of analytic conics for vertices, then determines valid edge segments by parameter search and dominance checks, with $O(n^4)$ worst-case complexity but practical reductions via planarity and locality (Jung et al., 2023).

4. Applications in Optimal Transport, Fluid Simulation, and Materials Science

Generalized Laguerre cells provide the geometric substrate for partial optimal transport problems, especially in particle-based free-surface fluid simulation and deformation mechanics. Given a target measure $m_i$ per site, one seeks the weights $\psi_i$ such that $|V_i^g| = m_i$, maximizing the Kantorovich functional

$$K(\psi) = \sum_{i=1}^n \int_{V_i^g(\psi)} (\|x-x_i\|^2 - \psi_i) dx + \sum_{i=1}^n \psi_i m_i.$$

A Newton or quasi-Newton solver updates $\psi$ using explicitly computed gradients and Hessians, which depend on facet and spherical patch areas (Plateau--Holleville et al., 9 Jan 2026). This analytic construction allows the enforcement of hard volume constraints with subcell precision in dynamic fluid simulation and supports the evaluation of forces (pressure, viscosity, surface tension) by direct integration.

In microstructure modeling, generalized balanced power diagrams provide highly flexible tessellations for representing non-convex or anisotropic grains, with curved (elliptic, hyperbolic, parabolic) interfaces that more accurately capture empirical morphologies than classical convex cells (Jung et al., 2023).

5. Generalizations: Metric and Algebraic Extensions

Classical Laguerre cells arise from quadratic, isotropic metrics. The GBPD extends this to arbitrary symmetric positive-definite $A_i$, inducing ellipsoidal growth and conic-section bisectors. Curved, non-convex cell boundaries result, and the analytic approach handles both the parametric and intersectional geometry (Jung et al., 2023).

In representation theory, analogues of Laguerre cells appear as explicit solutions to systems of coupled ODEs and recurrence relations in indecomposable $sl(2)$ modules. The “Laguerre-type cells” $\omega_{n,\beta}(x)$, defined via chain indices $\beta$, generalize Laguerre polynomials to modules with Jordan-block structure or diagonal Cartan action. The top chain reduces to $L_n^{(\alpha)}(x)$ (the classical Laguerre polynomial), with lower chains satisfying inhomogeneous differential equations and higher-order scalar ODEs. Unlike classical orthogonality, these functions admit bi-orthogonality relations but maintain simple, real zeros on $(0, \infty)$ (Bertrand et al., 2023).

6. Computational and Visualization Frameworks

The analytic geometric construction of generalized Laguerre cells enables direct, high-precision rendering and geometric queries in simulation pipelines. Key features of the dedicated volumetric renderer include:

• Free-surface extraction using impostors for spherical patches (represented as screen-space billboards at $x_i$).
• Fragment-level ray-testing via analytic sphere/intersection with projected facets.
• Interior ray traversal for volume effects, performed via facet-walking with no need for auxiliary acceleration structures.
• Meta-ball style normal smoothing, using weighted sums of normalized vectors to ensure visual continuity at cell boundaries (Plateau--Holleville et al., 9 Jan 2026).

These techniques yield a meshless, artefact-free rendering of complex fluid surfaces and internal structures, supporting advanced visualization requirements in scientific computing.

7. Comparison with Classical Laguerre and Related Structures

A comparison of key properties is summarized below:

Structure	Metric/Boundary	Cell Shape	Orthogonality	Applications
Classical Laguerre (Power) Cell	Euclidean, linear	Convex polytope	Yes (standard)	Mesh generation, OT, fluids
Generalized Laguerre Cell (Ball-clip)	Euclidean + sphere	Polytope ∩ ball	No (bi-orthogonality)	Free-surface fluids
GBPD (Anisotropic/elliptic metric)	Ellipsoidal, conic	Non-convex, curved	No (bi-orthogonality)	Anisotropic materials, grains
Indecomposable $sl(2)$ Laguerre cell	Algebraic	Polynomial function	No (bi-orthogonality)	Rep. theory, special function

A plausible implication is that generalized Laguerre cells provide a unifying geometric and analytic framework, subsuming classical constructions and extending their applicability to modern simulation and mathematical modeling problems characterized by anisotropy, domain restriction, and algebraic structure.

References: (Bertrand et al., 2023, Plateau--Holleville et al., 9 Jan 2026, Jung et al., 2023)