Smooth Polar Splines: Theory & Applications

• Smooth Polar Splines are spline functions defined over polar domains that enforce high-order smoothness at the origin using specialized algebraic constraints.
• They employ extraction matrices and projected polar harmonics to adapt standard tensor-product B-splines, ensuring continuity and stability in numerical simulations.
• These splines support isogeometric analysis and structure-preserving discretizations, enabling exact representations in geometric modeling of disk-shaped domains.

A smooth polar spline is a spline function defined over a domain with a central coordinate singularity—typically the origin in polar coordinates—that is constructed to achieve a desired degree of smoothness (commonly $C^1$ or $C^\infty$) at the singular point while maintaining standard spline properties elsewhere. Smooth polar splines adjust the basis and algebraic constraints of tensor-product B-splines or NURBS so as to compensate for the failure of standard B-splines to ensure regularity at the pole, enabling exact, stable, and efficient discretizations on disk-like or polar-parameterized geometries. They play a central role in geometric modeling, isogeometric analysis, and structure-preserving numerical PDE methods on domains with polar singularities or corners.

1. Mathematical Structure of Smooth Polar Splines

Standard tensor-product B-spline and NURBS constructions, when directly mapped via a polar coordinate transformation, exhibit a loss of regularity at the pole $r=0$, manifesting as discontinuities or kinked derivatives, which are not compatible with smooth physical fields or geometrical surfaces. Smooth polar splines are constructed specifically to enforce the necessary algebraic and differential regularity conditions at the pole, while retaining local support, partition of unity, and the capability for adaptive refinement.

The essential construction involves:

• Starting with a univariate B-spline (or NURBS) basis in the radial and angular directions.
• Characterizing and imposing necessary algebraic constraints on the spline coefficients near the pole, so that smoothness conditions are satisfied.
• Replacing the basis functions at the pole (i.e., those supported on the innermost "ring" of the radial coordinate) by specific linear combinations that correspond to regular polar or Cartesian harmonics at $r=0$.
• Ensuring that away from the pole, the basis coincides with the standard tensor-product B-spline basis.

For example, a $C^k$ condition at $r=0$ is equivalent to enforcing that only harmonics $r^\ell \cos(m\theta)$, $r^\ell \sin(m\theta)$ with $|m| \leq \ell \leq k$ and $|m| \equiv \ell \bmod 2$ are present in the center (pole) representation (jiang et al., 25 Jan 2026, Güçlü et al., 21 May 2025). The result is a function space isomorphic to the subspace of polynomials with required regularity at the origin.

2. Basis Construction and Extraction Operators

The construction of smooth polar splines proceeds via:

• Extraction matrices: Local extraction matrices are used to combine (possibly multi-degree) NURBS or B-spline basis functions to form $C^1$ or higher-order smooth bases with polar singularities. These matrices enforce algebraic constraints at the joins between patches or rings and at the pole, guaranteeing continuity and the necessary Hermite conditions for derivatives (Speleers et al., 2020).
• Projected polar harmonics: In the high-order framework, smooth polar splines are defined as the Galerkin $L^2$ projection of harmonic polar functions $S_\ell^m(r,\theta) = r^\ell e^{im\theta}$ onto the innermost tensor-product B-spline space. The construction yields a new collection of origin-centered B-spline basis functions ("polar splines") expressed as linear combinations of standard B-spline products in $(r,\theta)$, with explicit prolongation and restriction operators mapping between the standard and smooth subspaces (jiang et al., 25 Jan 2026).
• Dimension reduction: The replacement of $n_\theta$ linearly independent but singular basis functions supported at $r=0$ with only $(p+1)(p+2)/2$ smooth basis functions (for degree $p$) ensures a minimal, regular basis. This basis exactly reproduces the desirable regularity imposed by the underlying geometry (jiang et al., 25 Jan 2026, Güçlü et al., 21 May 2025).

A representative summary of basis modifications for $C^k$ polar splines appears in Table 1.

Standard basis near $r=0$	Smooth polar replacement	Dimension
$n_\theta$ splines $B_0(r)M_j(\theta)$	$(p+1)(p+2)/2$ projected polar harmonics	$(p+1)(p+2)/2$

3. Regularity Constraints and Local Projections

Enforcing regularity at the pole is achieved algebraically:

• $C^0$: All coefficients on the innermost ring are taken equal, replacing the set by their mean, ensuring well-posedness at the pole (Güçlü et al., 21 May 2025, Apel et al., 15 May 2025).
• $C^1$ and higher: Additional derivative-matching constraints are imposed, such as setting the first radial derivative to vanish at $r=0$, or more generally, requiring the reconstruction at $r=0$ to respect the structure of polar harmonics up to the desired order (Speleers et al., 2020, jiang et al., 25 Jan 2026).
• Projection operators: Local, matrix-free projection operators act only on the first "polar rings" of the coefficient array, replacing the original singular basis functions with the regularized ones while leaving the rest of the spline basis unchanged (Güçlü et al., 21 May 2025).

These operations are stable, dimension-preserving, and preserve compatibility with key structure-preserving discretization frameworks, including the de Rham complex in finite element exterior calculus—a property verified by commutation of the projections with differential operators (Güçlü et al., 21 May 2025).

4. Polar Spline Spaces and Geometric Modeling Capabilities

Smooth polar splines span a function space of physically and geometrically meaningful functions, permitting:

• Exact representation of classical conic sections and higher-degree polar objects: The extraction-matrix framework admits exact modeling of circles, ellipses, ellipsoids, and their generalized polar analogues (Speleers et al., 2020).
• Compatibility with multi-patch and polyhedral domains: The global space is constructed by gluing local polar spline patches via C1 (or higher) constraints across seams and at the central pole; blending functions associated with each control-net vertex are derived to ensure optimal smoothness (Mishra et al., 2023).
• Support for refinements and adaptivity: The spline-extraction and refinement operators (degree elevation, knot insertion) extend to polar splines, enabling the use of isogeometric analysis (IGA) techniques that require local adaptivity without sacrificing smoothness near corners or poles (Speleers et al., 2020, Apel et al., 15 May 2025).

5. Applications in Analysis, PDEs, and Simulation

Smooth polar splines are exploited in several key settings:

• Isogeometric Analysis: Regular polar splines ensure optimal convergence rates for Galerkin approximations on domains with polar singularities or corners, especially when used in conjunction with graded mesh refinement towards the singular point (Apel et al., 15 May 2025). Optimal approximation and discretization error rates are obtained provided appropriate grading and projection operators are used.
• Structure-preserving discretizations: In the FEEC framework, smooth polar splines support stable and structure-preserving discretizations of the de Rham sequence on disks and polar domains, with commuting projection operators and explicit basis functions for scalar, vector, and higher differential forms (Güçlü et al., 21 May 2025).
• Particle-in-cell (PIC) and spectral methods: High-order smooth polar splines dramatically improve the conditioning of mass and stiffness matrices, filter out unphysical spurious eigenmodes in eigenvalue problems, and suppress statistical noise in particle-based simulations by restricting the number of admissible Fourier modes near the pole (jiang et al., 25 Jan 2026).

6. Implementation and Numerical Behavior

Algorithmic realization of smooth polar splines rests on:

• Matrix assembly: All necessary extraction matrices and projection operators involve only local, sparse linear algebra and are compatible with standard NURBS-/B-spline data structures and refinement techniques (Speleers et al., 2020, Güçlü et al., 21 May 2025).
• Numerical stability: Maximal errors and convergence rates for systems such as Poisson or heat equations solved on polar-patched domains show no loss in accuracy or artifacts at the pole. Stiffness and mass matrices for the polar spline subspaces exhibit improved condition numbers under Jacobi or diagonal preconditioning, compared to their tensor-product analogues (jiang et al., 25 Jan 2026, Mishra et al., 2023).
• Software compatibility: Since all polar modifications are implemented via extraction matrices, existing CAD/CAE systems supporting NURBS or Bézier extraction can directly ingest polar splines (Speleers et al., 2020).

7. Connections, Limitations, and Extensions

Smooth polar splines generalize to a range of settings:

• Corners and graded refinement: A polar parameterization combined with graded mesh refinement enables optimal convergence in domains with corners or re-entrant singularities (Apel et al., 15 May 2025).
• Arbitrary valence polar caps: Polyhedral spline methods generalize to fans with arbitrary valence, handling $n$-gon topologies (Mishra et al., 2023).
• High-order regularity: $C^k$ and $C^\infty$ polar splines can be constructed up to the maximal order permitted by the B-spline degree and angular resolution, with precise algebraic characterization of the basis (jiang et al., 25 Jan 2026).
• Limitations: Exact $C^\infty$ regularity at the pole is approached asymptotically as the angular discretization resolves all polar harmonics; for finite $\Delta\theta$, a finite number of harmonics are retained (jiang et al., 25 Jan 2026). The approach is compatible with both rational and polynomial spline spaces.

A plausible implication is that smooth polar splines can serve as a universal regularizing framework for isogeometric analysis and geometric modeling on singular domains, with minimal disruption to existing spline technologies and significant benefits in numerical stability and fidelity.

References:

• (Speleers et al., 2020) A general class of $C^1$ smooth rational splines: Application to construction of exact ellipses and ellipsoids
• (Mishra et al., 2023) Polyhedral Splines for Analysis
• (Güçlü et al., 21 May 2025) A broken-FEEC framework for structure-preserving discretizations of polar domains with tensor-product splines
• (jiang et al., 25 Jan 2026) Smooth Polar B-Splines with High-Order Regularity at the Origin
• (Apel et al., 15 May 2025) Error Estimates and Graded Mesh Refinement for Isogeometric Analysis on Polar Domains with Corners