A regularization method for planar offset curves and bi-offset recognition

Rosanna Campagna; Salvatore Mondrone; Tomas Sauer

arxiv: 2605.18096 · v1 · pith:ELERMWISnew · submitted 2026-05-18 · 🧮 math.NA · cs.NA

A regularization method for planar offset curves and bi-offset recognition

Rosanna Campagna , Salvatore Mondrone , Tomas Sauer This is my paper

Pith reviewed 2026-05-20 00:29 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords offset curvesbi-offsetHermite splinespenalized regressioncenter line reconstructionplanar trajectoriescurvaturetrajectory smoothing

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The pith

Penalized Hermite splines regularize trajectories to produce bi-offset curves that best fit the original and reconstruct the center line from boundaries.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a two-stage method to create accurate offset curves for planar trajectories used in CNC machining and autonomous driving. In the first stage, penalized Hermite spline regression approximates the curve by fitting both positions and tangents while regularizing second derivatives to reduce jerk and singularities. In the second stage, two offset curves are constructed through simultaneous approximation of function values and derivatives. A mathematical model then defines the bi-offset as the version most consistent with the original generator curve and explicitly relates the offset range to pointwise curvature values. The central focus is using this model for adaptive reconstruction of the center line from external boundaries, supported by numerical tests showing reliability at each step.

Core claim

By first approximating trajectories with penalized Hermite splines that fit both positions and tangents under regularization on second derivatives, the method constructs offset curves through joint approximation of function values and derivatives. This leads to a bi-offset defined as the most fitting with the generator curve, where the offset range is related to pointwise curvature, enabling the adaptive reconstruction of the center line from external boundaries.

What carries the argument

The bi-offset model, obtained after penalized Hermite spline regularization of the generator curve and simultaneous approximation of values and derivatives to form the offsets, which then relates offset range to local curvature to recover the center line from boundaries.

Load-bearing premise

The first-stage penalized Hermite spline regression simultaneously fits positions and tangents while mitigating singularities and jerk sufficiently for the subsequent geometric offset construction to remain accurate and well-behaved without introducing new artifacts that would invalidate the bi-offset model.

What would settle it

A test case with a known exact generator curve and center line where the bi-offset reconstructed center line deviates from the original beyond the expected approximation error, or where the curvature-offset relation fails to predict the best fit, would falsify the model.

Figures

Figures reproduced from arXiv: 2605.18096 by Rosanna Campagna, Salvatore Mondrone, Tomas Sauer.

**Figure 2.** Figure 2: Base curve (black ’•−’), computed offset at distance τ = 0.5 (red ’- -’), interior bi-offset (blue ’· -’), exterior bi-offset (green ’· -’). in which the offsetting process is reversible. In order to conduct a pointwise analysis of the proposed algorithm, it is appropriate to decompose the overall procedure into a sequence of well-defined steps. At each step, a regression problem must be formulated, preced… view at source ↗

**Figure 3.** Figure 3: Base curve (blue ’•−’) and theoretical offsets, interior offset (magenta ’- -’) and exterior offset (green ’- -’), for different τ : top left τ = 0.1, right τ = 0.3; bottom left τ = 0.5, right τ = 0.7. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Base curve (black ’ [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗

**Figure 5.** Figure 5: Case τ = 0.3: interior (magenta ’- -’) and exterior (green ’- -’) theoretical offsets (a), OP-splines (red ’- -’) (b). allows to manage the boundary effects arising in the bi-offset approximations. Numerical results confirming these improvements are detailed in the tables of the next Test section. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗

**Figure 6.** Figure 6: Case τ = 0.3, σ = 1.0e − 2: interior (magenta ’- -’) and exterior (green ’- -’) theoretical offsets (a), OP-splines (red ’- -’) (b). 5.3. Test 3 (model comparison) This section presents some results on a comparison between the bi-offset approximations obtained both using our approach and different spline models, in the approximation of the dataset (in Step 1). Particularly we com21 [PITH_FULL_IMAGE:figu… view at source ↗

**Figure 7.** Figure 7: Case τ = 0.3: interior (or from above) (blue ’- -’) and exterior (or from below) (green ’- -’) BP-spline with refinement (a); interior (or from above) (blue ’- -’) and exterior (or from below) BP-spline without refinement (b). pare the results obtained by TP-spline, with the ones given by the classical P-spline [4], computed by a proprietary code, and the interpolating cubic spline, by the MATLAB function … view at source ↗

read the original abstract

Offset curves for planar trajectories are interesting in the generation of tool paths for numerically controlled industrial machines and in trajectory planning methods for autonomous driving systems. Theoretical offset curves may exhibit peculiar singularities, including self-intersections, which limit their use in practical applications. Existing approaches address these issue through geometric filtering techniques to detect and remove undesirable features but the computation of accurate and well-behaved offset curves remains a challenging task. We assume a first stage of functional approximation of trajectories by penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect, which is particularly relevant in motion planning and path smoothing applications. Then, taking into account the geometrical pointwise properties of the resulting curve, we design two offset curves through the simultaneous approximation of function values and derivatives. Then, a mathematical model to obtain the so-called bi-offset as most fitting as with the original generator curve is proposed, also relating the offset range and pointwise curvature values. The adaptive reconstruction of the center line from the external boundaries is a topic of interest and is the main focus of our work. Numerical experiments confirm the reliability of our approach at every stage of the resolution process.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper outlines a two-stage spline regularization for planar offset curves that adds a curvature-linked bi-offset model and center-line reconstruction, which is a coherent practical extension but not a big theoretical leap.

read the letter

The main takeaway is a staged method that first fits a trajectory with penalized Hermite splines to match positions and tangents while penalizing second derivatives to cut jerk, then builds offset curves by approximating both values and derivatives at once, and finally proposes a bi-offset model that ties offset distance to pointwise curvature for better agreement with the original curve, with the adaptive center-line recovery from the boundaries as the central goal. This targets real issues in CNC tool paths and autonomous trajectory planning where self-intersections and smoothness are problems. The simultaneous fitting of values and derivatives across stages gives some consistency, and linking the bi-offset to curvature is a straightforward geometric step that could help in practice. Numerical experiments are claimed to back each part, which is the right kind of evidence for this sort of applied work. The approach extends existing spline and offset techniques in a combined pipeline rather than starting from scratch, so the advance is incremental but targeted. On the softer side, the penalty parameter for the second derivatives is a free choice that will need case-by-case tuning, and without seeing the actual error tables, convergence rates, or head-to-head comparisons against standard geometric filters it is difficult to judge how large the practical gain really is. The assumption that the initial spline stage avoids new artifacts in the offsets seems reasonable from the description but would be stronger with more analysis of propagation. This is aimed at numerical analysts and engineers who need workable algorithms for smoothing offsets rather than readers hunting for deep new theory. It deserves a serious referee because the motivation is clear, the steps are logically connected, and the problem area is active enough that feedback on validation and comparisons would be useful.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a two-stage method for generating regularized planar offset curves. Stage one performs penalized Hermite spline regression that simultaneously fits positions and first derivatives while penalizing second derivatives to control jerk. Stage two constructs inner and outer offset curves from the resulting spline using pointwise curvature, introduces a bi-offset model that relates offset distance to curvature values, and develops an adaptive procedure to reconstruct the center line from the pair of external boundaries. Numerical experiments are reported to support reliability at each stage.

Significance. If the derivations and experiments hold, the work offers a direct regularization route to singularity-free offsets that integrates spline approximation with geometric offset construction. This could be useful for CNC tool-path generation and autonomous-vehicle trajectory planning, where jerk control and center-line recovery from boundaries are recurring requirements. The explicit linkage between offset range and pointwise curvature in the bi-offset model is a potentially distinctive element.

major comments (2)

[§4] §4 (bi-offset construction): the claim that the bi-offset is the 'most fitting' with the generator curve rests on a pointwise curvature relation; however, the manuscript does not appear to supply an error bound or stability estimate showing that the penalized spline's second-derivative control propagates to bounded curvature error in the offset step. Without this, it is unclear whether the subsequent center-line reconstruction remains accurate when the original curve has regions of high curvature variation.
[§3.2] §3.2 (Hermite spline regression): the penalty term is stated to mitigate jerk while preserving tangent fitting, yet the numerical section does not report the condition number of the resulting linear system or the sensitivity of the offset curves to the single free penalty parameter. This leaves open whether the method is robust across different trajectory lengths or sampling densities.

minor comments (3)

The abstract and introduction use 'bi-offset recognition' without a concise definition; a one-sentence statement of what quantity is being recognized would improve readability.
[Table 1] Table 1 (numerical results): the reported maximum deviation values lack units or normalization relative to curve length; adding this information would make the reliability claims easier to interpret.
[Figure 3] Figure 3 caption: the distinction between the reconstructed center line and the true medial axis is not stated; a brief clarification would prevent misinterpretation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4 (bi-offset construction): the claim that the bi-offset is the 'most fitting' with the generator curve rests on a pointwise curvature relation; however, the manuscript does not appear to supply an error bound or stability estimate showing that the penalized spline's second-derivative control propagates to bounded curvature error in the offset step. Without this, it is unclear whether the subsequent center-line reconstruction remains accurate when the original curve has regions of high curvature variation.

Authors: We agree that the manuscript does not supply a formal error bound or stability estimate demonstrating how the second-derivative penalty propagates to bounded curvature error in the offset construction. The bi-offset is defined via the pointwise curvature relation to achieve geometric fidelity with the generator curve. Numerical experiments in the paper, including cases with high curvature variation, show that center-line reconstruction remains accurate in practice. In the revision we will add a brief remark in §4 linking the penalty term to implicit control of curvature variation through the spline smoothness, thereby supporting stability of the reconstruction. This is a partial revision. revision: partial
Referee: [§3.2] §3.2 (Hermite spline regression): the penalty term is stated to mitigate jerk while preserving tangent fitting, yet the numerical section does not report the condition number of the resulting linear system or the sensitivity of the offset curves to the single free penalty parameter. This leaves open whether the method is robust across different trajectory lengths or sampling densities.

Authors: We agree that the absence of condition-number reporting and sensitivity analysis leaves the robustness claim incompletely supported. The penalty is introduced precisely to control jerk while the Hermite conditions preserve tangent fitting. In the revised version we will augment the numerical section with explicit condition numbers of the linear system for representative penalty values and sampling densities, together with plots or tables illustrating the sensitivity of the resulting offset curves to the penalty parameter across trajectories of different lengths. These additions will be included in the next manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper describes a constructive two-stage procedure: penalized Hermite spline regression to fit positions and tangents while regularizing second derivatives, followed by geometric construction of offset curves that incorporate pointwise curvature to define a bi-offset model and adaptive center-line reconstruction. No equations or steps in the abstract reduce by construction to the inputs (e.g., no fitted parameter renamed as a prediction, no self-definition of the bi-offset via its own outputs, and no load-bearing self-citation chains). The method is presented as building on standard spline regression and geometric offset properties, with numerical experiments cited for validation rather than internal self-consistency loops. This is a normal non-circular engineering proposal.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The approach rests on the assumption that penalized spline regression can be applied to trajectories without destroying the geometric information needed for offset construction.

free parameters (1)

penalty parameter for second derivatives
Introduced in the penalized Hermite spline regression stage to control smoothness and mitigate jerk; value not specified in abstract.

axioms (1)

domain assumption Trajectories admit a functional approximation by penalized Hermite splines that simultaneously fits positions and tangents while preserving sufficient geometric properties for offset curve construction.
Invoked as the first stage of the method in the abstract.

pith-pipeline@v0.9.0 · 5743 in / 1442 out tokens · 46430 ms · 2026-05-20T00:29:21.349681+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

penalized Hermite spline regression enabling the simultaneous fitting of positions and tangents. The regularization is imposed on the second derivatives, effectively mitigating the jerk effect... mathematical model to obtain the so-called bi-offset... relating the offset range and pointwise curvature values
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

offset curve generated by r ... ro(t) = r(t) + τ N(t) ... k = r'' · J r' / ||r'||^3 ... cusp when k(tc) = −1/τ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

(2021) Penalized hyperbolic-polynomial splines,Applied Mathematics Letters, 118

Campagna, R., Conti, C. (2021) Penalized hyperbolic-polynomial splines,Applied Mathematics Letters, 118

work page 2021
[2]

(2025) A Hermite spline model for data regression,Mathematics and Computers in Simulation, 229, pp

Campagna, R., Cotronei, M., Fazzino, D. (2025) A Hermite spline model for data regression,Mathematics and Computers in Simulation, 229, pp. 222 - 234

work page 2025
[3]

(2014) Properties of generalized offset curves and surfaces.Journal of Applied Mathematics, 2014:124240

Xuejuan Chen and Qun Lin. (2014) Properties of generalized offset curves and surfaces.Journal of Applied Mathematics, 2014:124240

work page 2014
[4]

and Marx Brian D

Eilers Paul H.C. and Marx Brian D. (1986) Flexible smoothing with b-splines and penalties.Statistical Science, Vol. 11, No. 2, 89-121

work page 1986
[5]

and Neff, C.A

Farouki, R.T. and Neff, C.A. (1990) Analytic properties of plane offset curves.Computer Aided Geometric Design, 7(1):83–99

work page 1990
[6]

and Neff, C.A

Farouki, R.T. and Neff, C.A. (1990) Algebraic properties of plane offset curves.Computer Aided Geometric Design, 7(1):101–127

work page 1990
[7]

and Salamon S

Gray, A., Abbena, E. and Salamon S. (2006) Modern differential geom- etry of curves and surfaces with mathematica,Chapman & Hall/CRC

work page 2006
[8]

17, Issue 2, 77-82

Hoschek, J (1985) Offset curves in the plane.Computer-Aided Design, Vol. 17, Issue 2, 77-82

work page 1985
[9]

Hoschek, J (1987) Approximate conversion of spline curves.Computer Aided Geometric Design, 4(1):59–66

work page 1987
[10]

5, Issue 1, 33-40

Hoschek, J (1988) Spline approximation of offset curves.Computer Aided Geometric Design, Vol. 5, Issue 1, 33-40

work page 1988
[11]

Hoschek, J (1988) Intrinsic parametrization for approximation.Com- puter Aided Geometric Design, 5(1):27–31. 25

work page 1988
[12]

Xuemin Hu, Long Chen, Bo Tang, Dongpu Cao, and Haibo He (2018) Dynamic path planning for autonomous driving on various roads with avoidanceofstaticandmovingobstacles.Mechanical Systems and Signal Processing, 100:482–500

work page 2018
[13]

Young Joon Ahn (2023) Hermite interpolation of offset curves of para- metric regular curves.AIMS Mathematics, 8(12): 31008-31021

work page 2023
[14]

Bai Li, Yakun Ouyang, Li Li, and Youmin Zhang (2022) Autonomous driving on curvy roads without reliance on Frenet frame: A cartesian- based trajectory planning method.IEEE Transactions on Intelligent Transportation Systems, 23(9):15729–15741

work page 2022
[15]

(2017)A Novel Method of Offset Approxima- tion along the Normal Direction with High Precision, Mathematical and Computational Applications, 22(3), 39

Zhao, H., Zhou, L. (2017)A Novel Method of Offset Approxima- tion along the Normal Direction with High Precision, Mathematical and Computational Applications, 22(3), 39. 26

work page 2017

[1] [1]

(2021) Penalized hyperbolic-polynomial splines,Applied Mathematics Letters, 118

Campagna, R., Conti, C. (2021) Penalized hyperbolic-polynomial splines,Applied Mathematics Letters, 118

work page 2021

[2] [2]

(2025) A Hermite spline model for data regression,Mathematics and Computers in Simulation, 229, pp

Campagna, R., Cotronei, M., Fazzino, D. (2025) A Hermite spline model for data regression,Mathematics and Computers in Simulation, 229, pp. 222 - 234

work page 2025

[3] [3]

(2014) Properties of generalized offset curves and surfaces.Journal of Applied Mathematics, 2014:124240

Xuejuan Chen and Qun Lin. (2014) Properties of generalized offset curves and surfaces.Journal of Applied Mathematics, 2014:124240

work page 2014

[4] [4]

and Marx Brian D

Eilers Paul H.C. and Marx Brian D. (1986) Flexible smoothing with b-splines and penalties.Statistical Science, Vol. 11, No. 2, 89-121

work page 1986

[5] [5]

and Neff, C.A

Farouki, R.T. and Neff, C.A. (1990) Analytic properties of plane offset curves.Computer Aided Geometric Design, 7(1):83–99

work page 1990

[6] [6]

and Neff, C.A

Farouki, R.T. and Neff, C.A. (1990) Algebraic properties of plane offset curves.Computer Aided Geometric Design, 7(1):101–127

work page 1990

[7] [7]

and Salamon S

Gray, A., Abbena, E. and Salamon S. (2006) Modern differential geom- etry of curves and surfaces with mathematica,Chapman & Hall/CRC

work page 2006

[8] [8]

17, Issue 2, 77-82

Hoschek, J (1985) Offset curves in the plane.Computer-Aided Design, Vol. 17, Issue 2, 77-82

work page 1985

[9] [9]

Hoschek, J (1987) Approximate conversion of spline curves.Computer Aided Geometric Design, 4(1):59–66

work page 1987

[10] [10]

5, Issue 1, 33-40

Hoschek, J (1988) Spline approximation of offset curves.Computer Aided Geometric Design, Vol. 5, Issue 1, 33-40

work page 1988

[11] [11]

Hoschek, J (1988) Intrinsic parametrization for approximation.Com- puter Aided Geometric Design, 5(1):27–31. 25

work page 1988

[12] [12]

Xuemin Hu, Long Chen, Bo Tang, Dongpu Cao, and Haibo He (2018) Dynamic path planning for autonomous driving on various roads with avoidanceofstaticandmovingobstacles.Mechanical Systems and Signal Processing, 100:482–500

work page 2018

[13] [13]

Young Joon Ahn (2023) Hermite interpolation of offset curves of para- metric regular curves.AIMS Mathematics, 8(12): 31008-31021

work page 2023

[14] [14]

Bai Li, Yakun Ouyang, Li Li, and Youmin Zhang (2022) Autonomous driving on curvy roads without reliance on Frenet frame: A cartesian- based trajectory planning method.IEEE Transactions on Intelligent Transportation Systems, 23(9):15729–15741

work page 2022

[15] [15]

(2017)A Novel Method of Offset Approxima- tion along the Normal Direction with High Precision, Mathematical and Computational Applications, 22(3), 39

Zhao, H., Zhou, L. (2017)A Novel Method of Offset Approxima- tion along the Normal Direction with High Precision, Mathematical and Computational Applications, 22(3), 39. 26

work page 2017