Quadratic B-spline Basis Functions

• Quadratic B-spline basis functions are piecewise quadratic polynomials with $C^1$ continuity, local support, and partition of unity properties, fundamental for constructing smooth curves and surfaces.
• These basis functions are widely applied in geometric modeling, finite element analysis (FEM), and isogeometric analysis (IGA) for discretizing PDEs and representing complex geometries.
• Applications range from cloth simulation and adaptive regression to constructing well-conditioned wavelet bases for efficient numerical methods.

Quadratic B-spline basis functions are piecewise-polynomial functions of degree two, central to geometric modeling, finite element analysis, and approximation theory. They are defined by a compact recurrence relation and possess key properties such as local support, non-negativity, partition of unity, and controllable smoothness. Their $C^1$ continuity (continuous first derivatives) makes them especially valuable for applications requiring smooth surface or curve representations, such as surface interpolation, cloth simulation, wavelet analysis, and isogeometric methods. The mathematical theory of quadratic B-splines has been widely extended to domains with irregular topology, multidimensional problems, and methods for adaptive approximation.

1. Mathematical Definition and Construction

Quadratic B-spline basis functions are a family of piecewise-quadratic polynomials constructed on a knot vector $\Xi = (\xi_0, \ldots, \xi_{n+2})$. The recursive Cox–de Boor formula is

$$N_{i,0}(x) = \begin{cases} 1, & \xi_i \le x < \xi_{i+1} \ 0, & \text{otherwise} \end{cases}$$

$$N_{i,2}(x) = \frac{x - \xi_i}{\xi_{i+2} - \xi_i} N_{i,1}(x) + \frac{\xi_{i+3} - x}{\xi_{i+3} - \xi_{i+1}} N_{i+1,1}(x)$$

In 2D, the tensor-product construction yields $N_{i,j}(u,v) = N_i(u) N_j(v)$. The basis can be generalized for bivariate domains using tensor products or explicit local constructions over triangulations.

Key properties:

• Local support: $N_{i,2}(x)$ is nonzero only over $[\xi_i, \xi_{i+3})$.
• Partition of unity: $\sum_i N_{i,2}(x) = 1$ for any $x$ in the domain.
• $C^1$ continuity: The basis and all spline functions in their span have continuous first derivatives at all knots (except those with multiplicity).

2. $C^1$ B-splines on Triangulations and Criss-Cross Meshes

Quadratic B-splines can be constructed on triangulations or criss-cross subdivisions, enabling $C^1$ splines on arbitrary planar meshes. The basis functions $B_{ij}(u,v)$ in the bivariate $C^1$ quadratic space $S_2^1(T_{mn})$ are localized around mesh vertices and defined to ensure $C^1$ continuity across adjacent elements.

Constructing such a basis involves:

• Expressing basis polynomials in barycentric coordinates on each triangle or local face.
• Imposing continuity by enforcing equality of values and directional derivatives at shared faces or edges.
• Merging (identifying) basis monomials according to these constraints, yielding explicit algebraic bases with minimal support (1409.3824).

This explicit construction applies to general topologies and is essential for finite element spaces, surface interpolation, and geometric modeling on arbitrary triangulations (1312.5533).

3. Curve and Surface Construction, Minimization Principles

Quadratic B-spline curves and surfaces are constructed as linear combinations of basis functions and control points: $\gamma(x) = \sum_i N_{i,2}(x)\, \mathbf{p}_i$

$$\mathbf{r}(u, v) = \sum_{i,j} N_{i,2}(u) N_{j,2}(v)\, \mathbf{p}_{ij}$$

For uniform, simple knots, these splines minimize a positive quadratic (moving least-squares) functional: $Q(p) = \sum_i W(x - x_i) [p - f(x_i)]^2,$ with $W$ matched to the B-spline's support. The B-spline interpolant at $x$ is the minimizer of $Q$, connecting spline theory to moving least squares and the wider field of approximation theory (1609.05313).

4. Applications: Finite Element Methods and Isogeometric Analysis

Quadratic B-spline bases are widely used for discretizing PDEs in finite element methods (FEM), particularly where $C^1$ smoothness is essential.

• Cloth simulation: A recent B-spline FEM using quadratic bases achieves globally $C^1$-continuous displacement fields, supporting consistent membrane and bending energy discretization. This property mitigates locking artifacts and mesh dependency, improving simulation realism and computational efficiency. A reduced integration quadrature scheme further accelerates computation while retaining accuracy (2506.18867).
• Burgers' Equation: Quadratic B-spline FEM for Burgers' equation delivers stable, accurate solutions with tridiagonal system matrices and efficient collocation/central difference time-stepping (2208.01092).
• Isogeometric analysis (IGA): Blended or SB-splines combine quadratic B-splines with Bernstein polynomials to handle irregular quadrilateral or hexahedral meshes, ensuring global $C^1$ smoothness and optimal PDE convergence rates on structured and semi-structured meshes (2111.04401).
• Mixed-element IGA: Quadratic B-splines generalize to mixed triangle-quad meshes by imposing $C^1$ trace and normal derivative conditions at interfaces, supporting advanced CAD and engineering geometries (2302.08278).

5. Extensions: Multi-degree and Adaptive Frameworks

Quadratic B-splines are a special case of multi-degree B-splines (MDB-splines), which allow the degree to vary per segment or region. MDB-splines are constructed using extraction operators that express basis functions as linear combinations of local B-spline segments. Transition functions, integral recurrences, and local convex combinations facilitate local refinement (knot insertion) and degree elevation, enabling fine-grained control in geometric modeling and engineering (1709.04998, 1809.01598).

Adaptive regression frameworks exploit quadratic B-splines for function estimation:

• LABS/MLABS models: Bayesian nonparametric regression expresses the target function as a sum (or tensor product) of B-splines of varying degree (including quadratic), randomly placed and indexed. The compact support and local smoothness of quadratic B-splines allow these models to adaptively capture smooth trends, sharp peaks, and discontinuities, while possessing theoretical guarantees of optimal approximation in Besov spaces (2101.12179, 2108.11863).

6. Wavelet Bases, Boundary Conditions, and Fast Algorithms

Quadratic B-spline bases serve as scaling functions for constructing wavelets with prescribed vanishing moments and boundary properties.

• Specially constructed quadratic spline-wavelet systems, built on B-spline scaling functions, provide short support and satisfy homogeneous Dirichlet boundary conditions, leading to sparse system matrices, low iterative solver costs, and uniform condition numbers in the discretization of elliptic PDEs. Their design ensures strict enforcement of boundary conditions, faster convergence, and computational superiority over longer-supported wavelets (1712.03643).

7. Analytical Formulas, Symbolic Properties, and Connections

Recent work derives explicit relations between quadratic B-spline and Bernstein basis functions, higher derivative formulas, and infinite series representations:

• For uniform quadratic B-splines,

$$N_{0,2}(w; p) = \sum_{j=0}^p (-1)^j (B_2(w-j) - B_1(w-j)),$$

relating B-spline values to translated Bernstein polynomials (2311.10124).

• Differential and Laplace transform approaches yield higher-order derivative and moment identities; p-adic integrals connect B-splines to special numbers like Bernoulli and Euler polynomials.
• Recurrence and functional equations revealed through this approach enhance symbolic-numeric computation and analysis.

Summary Table: Core Properties and Application Areas

Property / Trait	Description / Application	Reference
Local support, partition of unity	Stability, efficient computation, and local refinement	(1312.5533, 1609.05313)
$C^1$ smoothness ($C^1$ splines)	Accurate curvature modeling, critical for thin shells and cloth simulation	(2506.18867, 2111.04401)
Multi-degree, adaptivity	Local degree elevation, local refinement, and modeling flexibility	(1709.04998, 1809.01598)
FEM/PDE discretization	B-spline FEM for Burgers' eq, thin shells, and plate/shell bending	(2208.01092, 2506.18867)
Wavelet bases	Short-support wavelets for sparse, well-conditioned PDE solvers	(1712.03643)
Isogeometric analysis (IGA)	SB-splines and $C^1$ splines on unstructured meshes	(2111.04401, 2302.08278)
Explicit symbolic relationships	Formulas linking B-splines to Bernstein polynomials, series, and moments	(2311.10124)
Adaptive regression / LABS/MLABS	Bayesian methods for surfaces/functions of mixed smoothness	(2101.12179, 2108.11863)

Conclusion

Quadratic B-spline basis functions provide a unifying foundation for smooth curve and surface modeling, PDE discretization, and adaptive approximation. Their mathematical properties—$C^1$ continuity, local support, and stable partition of unity—underpin efficient algorithms and robust modeling in computational science and geometric design. The theory and practice of quadratic B-splines continue to evolve, with applications expanding into machine learning, isogeometric analysis, and symbolic computation.