Corotational Beam Spline

• Corotational beam spline is a flexible curve-fitting approach that models contours as equilibria of Euler–Bernoulli beams with compliant springs.
• It employs an energy minimization framework combined with finite-element formulations and dynamic re-indexing to ensure noise-robust contour reconstruction.
• Adaptive spring compliance provides uniform smoothing, preserving curvature features despite variable point densities and measurement inaccuracies.

A corotational beam spline (CBS) is a flexible curve-fitting and smoothing approach that models planar or spatial contours as the equilibrium configuration of a piecewise-linear Euler–Bernoulli beam supported at data points by compliant springs. Unlike standard spline or kernel-based smoothing, the CBS approach maintains a strong connection to the underlying continuum mechanics and provides rigorous control over smoothing through parameterized spring compliance. The method is noted for its rotationally objective formulation, dynamic re-indexing of sampled data, and adaptive smoothing that accounts for variable point density and measurement accuracy. CBS formulations have found application in geometric inverse problems such as apictorial jigsaw puzzle reconstruction and in isogeometric finite-element analysis of nonlinear beams (Orynyak et al., 8 Nov 2025, Choi et al., 19 Jul 2024).

1. Energy-Functional Foundation

The core of the CBS methodology is an energy minimization principle, where the configuration of the curve minimizes a sum of bending energy (from Euler–Bernoulli beam theory) and the potential energy stored in springs attached at the discrete data points: $$E_{\mathrm{total}}[W] = \frac{1}{2}\int_0^L (W''(t))^2\,\mathrm{d}t + \sum_{i=1}^{M+1} \frac{1}{2} D_i [ W(l_i) - \Pi_i ]^2$$ where $W(t)$ is the local transverse displacement, $M$ is the number of segments, $D_i$ is the spring rigidity at the $i$th point, and $\Pi_i$ is the signed data-point-to-segment distance.

The Euler–Lagrange equation yields a piecewise fourth-order differential equation with distributions at the supports: $$\frac{d^4 W}{dt^4} = \sum_{i=1}^{M+1} D_i ( W(l_i) - \Pi_i ) \delta( t-l_i )$$ Within each beam segment, $d^4W/dt^4 = 0$ holds, indicating local cubic polynomial displacements, with inter-segment jump conditions induced by the springs.

This continuous framework admits a direct connection to mechanical intuition: springs enforce fidelity to data, while beam bending regularizes high-frequency noise.

2. Finite-Element and Corotational Segment Representation

A CBS discretizes the contour into $M$ straight-line elements, each equipped with a local Frenet frame $(s_i, w_i)$ defined by the segment’s tangent $t_i$ and normal $n_i$. The state vector per segment is

$$Z_i(t) = \begin{pmatrix} W_i(t) \ \theta_i(t) \ M_i(t) \ Q_i(t) \end{pmatrix}$$

where $\theta_i$ is the cross-section rotation, $M_i$ the bending moment, and $Q_i$ the shear force. Local updates proceed via a transfer (or shape-function) matrix,

$Z_i(t) = p(t) Z_i(0), \quad 0 \leq t \leq l_i$

with cubic Hermite shape functions.

Corotational conjugation between segments ensures piecewise-rigid continuity and geometric exactness: $$\begin{aligned} W_{i+1,0} &= W_{i,1} \ \theta_{i+1,0} &= \theta_{i,1} - \psi_i \ M_{i+1,0} &= M_{i,1} \ Q_{i+1,0} &= Q_{i,1} - D_i ( W_{i,1} - \Pi_i ) \end{aligned}$$ where $\psi_i$ is the relative orientation angle between adjacent segments, computed by

$$\cos \psi_i = t_i \cdot t_{i+1}, \qquad \sin \psi_i = n_i \cdot t_{i+1}$$

This structure yields a sparse linear system with $8M$ unknowns and equations, enabling efficient solution by direct or iterative linear solvers.

3. Dynamic Re-Indexing for Contour Fidelity

A defining feature of CBS is dynamic re-indexing at each iteration. Given an approximate smooth spline $\mathcal{C}^{(k)}$, measured points $B_i$ are reprojected to their closest points $A_i^{(k)}$ on $\mathcal{C}^{(k)}$, and re-indexed by ordering along the curve’s arc-length. This procedure ensures that the beam elements connect consecutive data in physical order, preserving topology despite noise-induced irregular sampling or out-of-order input.

This dynamic association of support points with the evolving curve guarantees that data assignment accurately follows the geometric refinement, which is crucial for convergence and avoiding outlier-driven distortion.

4. Adaptive Spring Compliance and Uniform Smoothing

To realize genuine uniform smoothing along the contour irrespective of data sampling density, the springs’ rigidity $D_i$ is chosen proportional to the local arc-length half-span: $$D_i = \frac{L_i}{h^4}, \qquad L_i = \frac{ s_i + s_{i+1} }{2 }$$ with $s_i$ as the length of the adjacent segment and $h$ the principal smoothing parameter (bandwidth).

This ensures that, in aggregate, every arc-length unit contributes equally to the total spring force, mitigating overconstraint in densely sampled regions and underconstraint elsewhere. For smoothing of externally-imposed sinusoidal perturbations with semi-period $\xi$, the CBS transfer function is

$$\text{Amplitude scaling} = \frac{ \xi^4 }{ h^4 + \xi^4 }$$

Features with $\xi \ll h/2$ are suppressed, while longer wavelengths are preserved, providing a physically interpretable, one-shot smoothing filter with a clear cutoff scale.

5. Iterative Optimization Procedure

CBS solution proceeds by an outer fixed-point iteration on the curve geometry:

1. Initialization: Select a coarse polygonal subsampling so that adjacent angles $\psi_i < 90^\circ$, set an initial conservative $h_0$.
2. Linear Solve: Assemble and solve the transfer-matrix/conjugation system for the current set of $D_i^{(k)}$ and data gaps.
3. Dynamic Re-Indexing: Project measured data to the updated spline, re-sort by curve order for correct topology.
4. Bandwidth Reduction: Decrease $h_{k+1} = h_k / 1.2$, iteratively sharpening the fit.
5. Convergence Check: Iterations stop once maximum change in projection points falls below tolerance or target $h$ is reached.

Each cycle solves a well-posed linear system; practical convergence is rapid—even with high noise—due to the physics-informed smoothing and dynamic data assignment.

6. Preservation of Curvature Features and Robust Contour Matching

By enforcing $D_i \propto L_i$, CBS ensures that local smoothing is dictated solely by measurement error, not by the density of sampled points. This prevents high-density clusters from dominating local spline behavior, preserving underlying geometric features such as curvature.

The geometric continuity enabled by the CBS normal orientation $n_i^\theta(t)$ supports second-order accurate, noise-robust curvature computations: $$\kappa(s) = \frac{ x' y'' - y' x'' }{ ( x'^2 + y'^2 )^{3/2} }$$ This forms the basis for contour side-matching via

$$E_{ab} = \int_0^{L} [ \kappa_a(s) - \kappa_b(s) ]^2 \, \mathrm{d}s$$

allowing for physically meaningful comparison of jigsaw puzzle contours even under non-uniform sampling or large measurement errors.

The CBS approach requires only a single, physically interpretable parameter ($h$), and does not depend on ad hoc kernel windows or multi-pass “fairing” algorithms.

7. Applications and Comparative Context

CBS was introduced with demonstrable success in the automatic assembly of apictorial jigsaw puzzles, accurately reconstructing all 54 pieces from noisy, digitized boundary data (Orynyak et al., 8 Nov 2025). Key practical metrics include robust curve reconstruction in the presence of high noise and variable point density, with results sensitive only to measurement accuracy.

In the broader context of beam analysis, related corotational beam-spline methodologies using isogeometric analysis and mixed finite element approaches also exist, as in the Cosserat rod formulation with B-spline interpolation for nonlinear elastodynamics (Choi et al., 19 Jul 2024). While both approaches share an underlying mechanical motivation and objectivity, CBS focuses on geometric inverse problems and contact/matching, utilizing dynamic data association and uniform smoothing, whereas isogeometric beam splines exploit mixed variational formulations, energy-momentum consistent time integration, and patch-wise approximation for physical fidelity and computational efficiency in nonlinear structural analysis.

CBS thus represents a rigorously grounded, robust, and parameter-minimal tool for physically-inspired contour fitting, data smoothing, and precise localization of curvature features in geometric reconstruction and matching tasks.