Spline Control Representations

• Spline control representations are a mathematical framework that parameterizes piecewise-polynomial functions using control coefficients and basis functions with guaranteed continuity and locality.
• They extend to various forms such as B-splines, Bézier, Hermite, and trigonometric splines, enabling efficient geometric modeling and precise control net manipulation.
• These techniques are widely applied in optimal control, machine learning, and scientific computing to reduce parameter complexity and improve computational stability.

Spline control representations constitute a foundational paradigm for parameterizing piecewise-polynomial functions using a finite set of “control” coefficients multiplied by basis functions with specified continuity. This framework encompasses univariate and multivariate splines (e.g., B-splines, Bézier splines, Hermite splines, trigonometric splines), extending into tensor products, polyhedral and simplex splines, and generalizes to time-dependent representations for optimal control, machine learning, and geometric modeling. The selection of basis (e.g., B-spline, cardinal, Bézier, trigonometric) determines locality, smoothness, and computational properties, while the structure of the control net governs shape manipulation, stability, and approximation.

1. Mathematical Foundation and Basis Construction

Spline control representations express a function $f$ as a weighted sum over basis functions: $f(x) = \sum_{i=1}^N c_i \psi_i(x)$ where $\{c_i\}$ are control coefficients (“control points”) and $\{\psi_i\}$ are a basis that possesses local support, partition of unity, and specified smoothness.

B-spline basis: Constructed via the Cox–de Boor recursion, B-splines of degree $d$ on knot vector $\{\tau_i\}$ satisfy $C^{d-1}$ continuity at the knots. In the uniform case: $$N_{i,0}(u) = \begin{cases} 1, & u\in[u_i,u_{i+1}) \ 0, & \text{otherwise} \end{cases}$$

$$N_{i,d}(u) = \frac{u-u_i}{u_{i+d}-u_i} N_{i,d-1}(u) + \frac{u_{i+d+1}-u}{u_{i+d+1}-u_{i+1}} N_{i+1,d-1}(u)$$

The curve is then $C(u) = \sum_i N_{i,d}(u) P_i$ for control points $P_i$ (Zhou, 2023).

Bernstein–Bézier form: A degree-$n$ Bézier curve is given by $C(t) = \sum_{i=0}^n \beta_i b_{i,n}(t)$, where $b_{i,n}(t)=\binom{n}{i}(1-t)^{n-i} t^i$. This form guarantees global $C^{n-1}$ continuity for single-segment splines, and can be concatenated with matching endpoint and derivative conditions for composite splines (Knodt, 2022).

Hermite splines: Hermite forms use endpoint values and derivatives as control variables. Cubic and quintic Hermite splines, for example, embed position and derivative data at breakpoints, achieved through Hermite basis functions ($H_1$ through $H_4$ for cubic; $h_0$ through $h_5$ for quintic), and can be efficiently mapped to Bézier control points for evaluation and differentiation (Kondo et al., 13 Nov 2025).

Trigonometric splines: Interpolation trigonometric splines may be expressed via: (a) Fourier coefficient expansion, (b) trigonometric B-spline bases (quasi-local with built-in $C^{r-1}$ smoothness), or (c) cardinal (fundamental) splines with global support. The basis form dictates localization and computational trade-offs (Denysiuk et al., 2024).

2. Properties of Control Nets and Locality

The control net (arrangement and semantics of control points) dictates the geometric and analytic behavior of the spline.

• Locality: The action of altering a single control coefficient primarily affects the function in a neighborhood determined by the support of the corresponding basis function. For uniform B-splines of degree $d$, each basis function is nonzero on $d+1$ knots, conferring strict local control (Zhou, 2023).
• Continuity: The regularity of the basis functions enforces $C^{d-1}$ continuity at knots for B-splines, $C^0$ and $C^1$ for cubic Bézier and Hermite, and $C^{r-1}$ for trigonometric B-splines.
• Shape Preservation: Cardinal (Catmull–Rom) and cubic Hermite splines utilize tangent vectors constructed from data points (e.g., $T_k = \tau(p_{k+1} - p_{k-1})$) to balance smoothness and fidelity, preventing spurious oscillations (e.g., Runge phenomenon) (Mukherjee, 2020, Arasteh et al., 2020).
• Convex Hull: Bézier splines and convex-combination B-spline bases guarantee that the spline curve lies within the convex hull of its control points, enhancing shape predictability (Stelia et al., 2017).

3. Advanced Control Structures: Polyhedral, Simplex, and Trivariate Nets

Control representations have been generalized beyond regular grids:

• Simplex splines on triangulations: On the Powell–Sabin 12-split, $C^3$ quintic simplex-spline bases $[1504.02628] [1505.01801]$ are constructed with knot-multiplicity codes, ensuring positive partition of unity, barycentric Marsden identities, and explicit $C^r$ join conditions across interfaces:
  • Domain points $\xi_i$ are defined as averages of dual factorizations of the Marsden identity.
  • Explicit algebraic formulas relate control points across neighboring triangles for $C^0$, $C^1$, and $C^2$ continuity.
• Bi-cubic polyhedral splines: Polyhedral control nets allow patchwise bi-cubic BB-form representations associated to vertices of arbitrary valence (grids, stars, $n$-gons, T-junctions, poles), enforcing $C^1$ (or $G^1$) continuity algebraically. Each global basis function collects the local BB-coefficients touching a single net vertex, with explicit $C^1$ matching conditions across patch boundaries (Mishra et al., 2023).
• Full trivariate representations: For volume domains, one uses tensor-product, hierarchical, T-spline, and locally refined (LR) B-spline models, each with distinct control mesh semantics and local refinement strategies (Dokken et al., 2018). For example:
  • Block-structured: Uniform grid, strictly global refinement.
  • THB: Hierarchically nested, truncated for partition of unity.
  • T-splines: Enable single-vertex (cell) refinement with T-junctions.
  • LR B-splines: Arbitrarily local, single-plane refinements.

4. Spline Controls Beyond Geometry: Machine Learning, Optimal Control, and Scientific Computing

Spline control representations have substantial impact as parameter-efficient, regularized function representations in scientific and engineering computation.

• Neural ODEs and DNN Controls: B-spline parametrization of network “controls” decouples the number of trainable parameters from the discretization depth. The continuous network "control" $u(t)$ is expanded as $u(t) = \sum_{l} c_l B_l^d(t)$, yielding smoother parameter trajectories, improved robustness to hyperparameters, and order-of-magnitude parameter reduction relative to layerwise training (Günther et al., 2021).
• Spline-interpolated control in trajectory optimization: Representing control inputs (e.g., in trajectory planners or in Model Predictive Path Integral Control) as spline functions reduces dimensionality and enforces $C^2$ smoothness, with the control coefficients optimized directly. This approach yields smoother trajectories under high exploration noise and simplifies time-discretization trade-offs (Miura et al., 2024, Kondo et al., 13 Nov 2025).
• Implicit spline segmentation in DL: Control grids predicted by CNNs define implicit bivariate spline functions, whose zero level-set yields the segmentation boundary. Spline control nets drastically reduce output parameterization vs. pixelwise classification, allow gradient backpropagation through the upsampling matrix, and guarantee smooth, topology-flexible boundaries (Barrowclough et al., 2021).
• Spline-based representation in phase-field models: Cubic spline fits to free energy curves achieve lower $L^2$ error, better capture of spinodal structure, and orders-of-magnitude reduced computational time compared to high-degree Redlich–Kister polynomials, due to local support and $C^2$ regularity of the control representation (Teichert et al., 2016).

5. Interconversion, Generalization, and Trade-offs

Spline control representations admit linear interconversion between basis types (e.g., Catmull–Rom $\leftrightarrow$ Bézier), and can be regularized or adapted as application needs dictate.

• Basis conversion: For cubic Catmull–Rom and Bézier segments, explicit linear transformations of the control points achieve segmentwise equivalence. The mapping is a $4 \times 4$ invertible matrix, allowing applications to pipeline sharing (e.g., between sensor-fusion and CAD) (Arasteh et al., 2020).
• Design flexibility: Spline controls generalize to trigonometric, Hermite, and polyhedral variants for greater expressiveness (e.g., periodic boundary conditions via trigonometric B-splines (Denysiuk et al., 2024); $C^d$ by using higher-degree Hermite splines; locally adapted mesh topologies).
• Properties table:

Representation	Control Net Structure	Main Properties
Uniform B-spline	Regular grid	Local support, $C^{p-1}$, partition of unity
Polyhedral spline	Generalized mesh (quads/etc)	$C^1$ on general meshes, local BB forms
Hermite	Knots with derivatives	Direct boundary derivative enforcement
Bézier	Polygonal chain (segments)	Convex hull, $C^0$/optionally $C^1$
Trigonometric	Equispaced nodes	$C^{r-1}$, global (via Fourier) or quasi-local
Neural ODE spline	Layer-independent coefficients	Smooth control, decoupled parameter count

The choice of control representation is determined by requirements for continuity, locality, parametric efficiency, numerical conditioning, regularity (e.g., needed for PDE discretization), and interactivity (CAD/CAGD). Modern research continues to expand the range of admissible control structures and purposes, from physics-based simulation to deep learning, each leveraging the core spline control framework for expressive, tractable, and robust function representation.