PCC Elements in Discrete Geometry

• PCC elements are discretization tools that partition manifolds into cells with strictly constant sectional curvature, enabling precise geometric modeling and simulation.
• They underpin discrete exterior calculus by allowing explicit curvature assignment and the definition of robust numerical operators for various model spaces.
• Practical applications include computer graphics, continuum robotics, and Regge calculus, though their assumption of constant curvature may limit accuracy in heterogeneous conditions.

Piecewise Constant Curvature (PCC) elements are a critical concept in discrete and computational differential geometry, geometric modeling, and continuum robotics. In the PCC paradigm, a complex manifold or geometric object is discretized into a finite set of cells, each endowed with a metric of strictly constant sectional curvature—zero in the piecewise-flat case and nonzero (positive or negative) in the truly PCC case. This enables tractable representation, explicit curvature assignment, and robust discretizations for geometric processing, simulation, and control.

1. Formal Definition and Mathematical Structure

In the context of a triangulated or cellulated manifold, a PCC discretization replaces the global smooth geometry with locally constant curvature elements. Each cell (typically a simplex, polygon, or polyhedron) is prescribed a metric of constant sectional curvature κ, allowing isometric embedding (up to congruence) into a model space of constant curvature—Euclidean (κ=0), hyperbolic (κ<0), or spherical (κ>0) (Glickenstein et al., 2014). For a surface or manifold (M, T), PCC data consist of:

• A triangulation (or, more generally, a cell decomposition) T of the underlying space M.
• For each cell (e.g., triangle), a constant curvature value κ, and edge length data compatible with isometric embedding in the corresponding model geometry.
• For higher dimensions, an analogous prescription applies to n-simplices.

The geometric structure is tightly controlled via compatibility conditions to ensure local realizability and global consistency. For example, in 2D, a triangle with edge lengths ℓ₁₂, ℓ₂₃, ℓ₃₁ can be realized as a spherical, hyperbolic, or Euclidean triangle according to the value of κ and triangle inequalities suitable for the model space (Glickenstein et al., 2014).

2. Discrete Exterior Calculus and Discrete Curvature Operators

A major theoretical development for PCC elements is the adaptation of discrete exterior calculus (DEC) from the piecewise-flat (PF) setting to the PCC regime (McDonald et al., 2012). In DEC, geometric operators—such as the exterior derivative, Hodge star, and curvature tensors—are discretized on hybrid domains built from the interplay of the triangulation and its dual structure (e.g., circumcentric duals). In PF settings, curvature is concentrated on hinges (codimension-2 simplices), with the Riemann tensor assigned as an eigen-decomposition:

${\bf Riem} = \hat{h}^* \cdot \frac{h}{A_h^*}$

where $h$ is the deficit angle at a hinge and $A_h^*$ is the dual area. For PCC elements, this is generalized by replacing the defect $h$ with a constant curvature $\kappa$ throughout the cell:

$${\bf Riem}_{\text{PCC}} = \hat{h}^* \cdot \frac{\kappa}{A_h^*}$$

This yields curvature operators that are constant on each cell, preserving the geometric interpretation of curvature as an integrated, cellwise quantity. The DEC formalism, with its solder/moment arm forms linking primal and dual complexes, remains applicable (McDonald et al., 2012).

3. Duality Structures, Discrete Conformal Variations, and Classification

The geometric and combinatorial richness of PCC elements is further illuminated by discrete conformal geometry frameworks, particularly in the context of triangulated surfaces (Glickenstein et al., 2014). Each edge is supplemented with a pair of "partial edge lengths" $d_{ij}$ and a duality structure that defines a center for each simplex, yielding orthogonal dual decompositions.

Discrete conformal structures are then generated through vertex-based scaling parameters $(f_i)$:

• The edge lengths $\ell_{ij}$ are functions of the $f_i$, giving
  • In Euclidean geometry: $\partial \ell_{ij} / \partial f_i = d_{ij}$
  • In hyperbolic: $\partial \ell_{ij} / \partial f_i = \tanh(d_{ij})$
  • In spherical: $\partial \ell_{ij} / \partial f_i = \tan(d_{ij})$

Angle variation formulas and closed-form expressions for curvatures at vertices enable variational principles, such as the existence of curvature functionals $F$ with $\partial F / \partial u_i = K_i$, where $K_i$ is the discrete curvature at vertex $i$. A complete classification of all such structures is provided by parameterizing $d_{ij}$ and $\ell_{ij}$ in terms of vertex and edge constants $\alpha$, $\eta$ (Glickenstein et al., 2014).

4. Curvature Concentration, Geodesics, and Combinatorial Formulas

In both Riemannian and Finsler settings, when each cell is flat (PCC with κ=0) or generally of constant curvature, all intrinsic curvature is concentrated at lower-dimensional features. For surfaces, the total discrete curvature at a vertex $x$ can be written, e.g., as

$K(x) = 2\pi - \sum \text{angles at } x$

with the angles computed with respect to the cell's local geometry (Xu et al., 2016). In Finsler geometry, the curvature may also depend on the direction (the "flag curvature"). The extension of geodesics through a vertex is characterized by explicit curvature constraints determined by the local combinatorics and metrics (Xu et al., 2016).

A combinatorial Gauss-Bonnet formula holds in several contexts, generalizing its smooth analog:

$\sum_{\text{vertices } x} K(x) = O_m \chi(M)$

where $O_m$ is the (constant) indicatrix length in a piecewise flat Landsberg surface and $\chi(M)$ is the Euler characteristic (Xu et al., 2016).

5. Discretizations for Geometry Processing and Numerical PDEs

PCC elements provide the foundation for advanced discretization schemes in computational geometry and numerical relativity. Methods based on hybrid cells and dual tessellations allow the definition of discrete Laplacians, curvature operators, and integral geometric quantities that respect intrinsic and extrinsic curvature (McDonald et al., 2012, Conboye, 2016). For instance, in the discretization of the Einstein–Hilbert action one obtains

$I_{EH} = \frac{c^4}{8\pi G} \sum_h h A_h$

with $h$ replaced by $\kappa$ in the PCC context, and $A_h$ being the measure of the hybrid cell. Volume-averaged or region-averaged curvature measures support mesh-independent definitions and are valid in arbitrary dimension (Conboye, 2016).

6. Practical Applications and Limitations

PCC discretizations underpin diverse applications:

• In general relativity and cosmology, spacetime is modeled as a collection of cells each solving Einstein's equations with constant curvature, relevant for Regge calculus and discrete Hamiltonian evolutions (McDonald et al., 2012).
• In computer graphics and geometric modeling, meshes composed of PCC elements support accurate surface processing, ensure geometric operator consistency, and permit discrete conformal mapping (Glickenstein et al., 2014).
• In robotics, piecewise constant/approximate curvature models are foundational to the modeling of continuum and soft robotic arms, with extensions for non-constant or affinely-varying curvature yielding improved tracking of deformations in contact or under load (Stella et al., 2022).

Limitations include the assumption of intrinsic constancy of curvature within each cell, which becomes inaccurate under heterogeneous loading, highly irregular actuation, or environments where curvature varies sharply over short scales. For such scenarios, piecewise-affine or higher-order discrete curvature models have been developed, offering up to 30% reduction in end-effector positioning error in soft robotics benchmarking (Stella et al., 2022).

7. Connections to Spline Approximation and Discrete Function Analysis

The approximation of smooth curves by piecewise constant (or linear) curvature curves is mathematically precise within Sobolev normed spaces, with error guarantees given in terms of Hausdorff and 1-Wasserstein distances (Gournay et al., 2019). The smoothing of discrete curves back to continuous ones is achieved using B-splines, with Eulerian numbers featuring as natural coefficients in the construction. This forms a rigorous analytical basis for the structure-preserving discretization of geometric objects in both theoretical and computational settings.

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In conclusion, piecewise constant curvature elements unify a broad range of discrete geometric modeling paradigms. They enable explicit curvature assignment, support variational and combinatorial formulations amenable to computation, and are flexible for both theoretical analysis and practical deployment in simulation, design, and physical control of complex geometric structures. Extensions to affine and higher-order curvature models directly address the intrinsic limitations of the PCC framework for scenarios with nonuniform or dynamically evolving curvature while maintaining computational tractability.