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arxiv: 2603.21831 · v2 · submitted 2026-03-23 · 💻 cs.RO · math.DG

Directional Mollification for Knot-Preserving C^(infty) Smoothing of Polygonal Chains with Explicit Curvature Bounds

Alfredo Gonz\'alez-Calvin , Juan F. Jim\'enez , H\'ector Garc\'ia de Marina This is my paper

Pith reviewed 2026-05-15 01:09 UTC · model grok-4.3

classification 💻 cs.RO math.DG
keywords directional mollificationpolygonal chain smoothingC^∞ curvescurvature boundsvertex preservationlocal supportgeometric modelingcurve approximation
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The pith

Directional mollification turns polygonal chains into C^∞ curves that intersect every original vertex exactly and carry closed-form curvature bounds.

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

The paper introduces a directional mollification operator that converts any polygonal chain into an infinitely differentiable curve approximant. The smoothed curve can be made arbitrarily close to the input pointwise and uniformly on compact sets while still passing exactly through each prescribed vertex. Unlike ordinary mollification, the directional version keeps local support, so editing one segment changes the output only nearby, and it supplies explicit curvature bounds. A single parametric family recovers both the new operator and classical mollification as special cases, giving a unified way to obtain smooth curves with exact knot interpolation.

Core claim

Starting from a polygonal chain through prescribed knots, the directional mollification operator produces C^∞ curve approximants that are arbitrarily close to the original curve pointwise and uniformly on compact subsets while still intersecting the original vertices. The construction admits local support, closed-form curvature bounds, and belongs to a parametric family that also contains conventional mollification.

What carries the argument

The directional mollification operator, a vertex-preserving variant of mollification that maintains local support and supplies explicit curvature expressions.

If this is right

  • Smoothed curves can serve directly as reference paths in robotics and CNC machining that must pass exactly through given waypoints.
  • Explicit curvature bounds let designers enforce turning-radius limits without post-processing the curve.
  • Local support allows incremental updates to long chains without recomputing the entire smoothed output.
  • The parametric family lets users choose any point between global smoothing and strict vertex preservation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Optimization routines that previously treated polygonal constraints as non-differentiable could replace them with these smooth interpolants while retaining hard waypoint fidelity.
  • Similar directional constructions may extend to closed loops or to surfaces for knot-preserving smoothing in 3-D modeling.
  • Analytic curvature formulas could support direct stability proofs for feedback controllers that track the resulting paths.

Load-bearing premise

A directional variant of mollification can be defined that preserves exact vertex intersections while achieving arbitrary closeness, local support, and closed-form curvature bounds simultaneously on arbitrary polygonal chains.

What would settle it

A concrete polygonal chain together with a chosen epsilon for which every C^∞ approximant that passes through all vertices either deviates by more than epsilon on some compact set or fails to admit a closed-form curvature bound.

Figures

Figures reproduced from arXiv: 2603.21831 by Alfredo Gonz\'alez-Calvin, H\'ector Garc\'ia de Marina, Juan F. Jim\'enez.

Figure 1
Figure 1. Figure 1: Example of mollification of a polygonal chain 𝑓 ∶ ℝ → ℝ2 , where the coordinates in ℝ2 are expressed as (𝑢, 𝑣). The green curve shows the (conventionally) mollified function 𝐹0.4 as in Theorem 2 with the mollifier presented in Example 1. The number 0.4, stands for a parameter 𝜀 = 0.4 in which the mollification depends. The original (black) curve is an illustrative case of a polygonal chain curve whose knot… view at source ↗
Figure 2
Figure 2. Figure 2: Representation of the conventional and directional mollification of the strictly increasing function 𝑓(𝑥) = 𝑥 ind(−∞,0)(𝑥) + 30𝑥 ind[0,∞)(𝑥) (left picture) and the function 𝑡 ∈ [0, 2𝜋] ↦ (2 + cos(2𝑡))(cos(𝑡),sin(𝑡)) ∈ ℝ2 (right picture, where the coordinates of ℝ2 are shown as (𝑢, 𝑣)), where in both cases the used mollifier 𝜑 is as in (1). For the latter, the convolution is carried out by extending its dom… view at source ↗
Figure 3
Figure 3. Figure 3: Representation of the conventional and directional mollification approaches for a 𝑝 = 6 polygonal chain in ℝ2 defined as in (2), where the coordinates of ℝ2 are expressed as (𝑢, 𝑣). The top left picture represents in black the original polygonal chain curve 𝑓, in dashed green its conventional mollification 𝐹0.4 , in red its directional mollification 𝐹̂ 0.4 , and as a blue transparent patch the convex hull … view at source ↗
Figure 4
Figure 4. Figure 4: Representation of a 𝑝 = 7 polygonal chain in ℝ3 , and the curvature and curvature bounds of the conventional and directional mollifications. The coordinates in ℝ3 are expressed as (𝑢, 𝑣, 𝑤). The left picture represents in black the polygonal chain curve 𝑓 confined in a unit volume cube in ℝ3 , shown in blue. It also represents its conventional 𝐹𝜀 and directional mollifications 𝐹̂ 𝜀 in dashed green and soli… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison and representation of a family of curves using (12), where the mollifier used is the one presented in Example 1 with 𝜀 = 0.4 and different approaches of polynomial splines. The left picture represents in black a 𝑝 = 6 polygonal chain curve in ℝ2 whose vertices are represented as black dots, and the components of ℝ2 as (𝑢, 𝑣). It also shows in blue a cubic spline, in red a B-spline, and in green … view at source ↗
read the original abstract

Starting from a polygonal chain (a first-order polynomial spline) through prescribed knots (vertices), we introduce the \textit{directional mollification} operator, which acts on polygonal chains and locally integrable functions, and produces $C^{\infty}$ curve approximants arbitrarily close -- pointwise and uniformly on compact subsets -- to the original curve, while still intersecting the original vertices. Unlike standard mollification, which confines the smoothed curve to the convex hull of the image of the original curve and does not preserve the vertices, the directional construction permits local and vertex-preserving smoothing. That is, modifying a single line segment from the polygonal chain alters the $C^{\infty}$ output only on that segment and within an explicitly controllable small neighborhood of its endpoints. The operator admits closed-form curvature bounds and yields infinitely differentiable curves with analytic control over curvature. We further develop a parametric family of smoothing operators that contains both the conventional mollification and the proposed directional variant as special cases, providing a unified geometric framework for converting non-differentiable polygonal data into smooth curves with exact point interpolation, computational simplicity, explicit curvature control, and strong local support properties. These features make the method directly useful for geometric modeling, curve design, and applications that require both smoothness and strict knot/waypoint fidelity, such as in robotics, computer graphics and CNC machining.

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.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces the directional mollification operator, which converts polygonal chains into C^∞ curves that exactly intersect the original vertices, achieve arbitrary uniform closeness on compact sets, maintain strict local support, and admit closed-form curvature bounds. It further defines a parametric family of kernels that unifies standard mollification and the directional variant as special cases.

Significance. If the construction and bounds hold, the work supplies a practical tool for converting non-smooth polygonal data into smooth curves with exact knot preservation and explicit curvature control. These properties are directly useful in robotics path planning, CNC machining, and geometric modeling, where standard mollification fails to preserve vertices or provide local support. The explicit integral expressions for curvature and the parametric unification are concrete strengths.

minor comments (3)
  1. [§3.1] §3.1: The definition of the directional kernel family (Eq. 8) introduces the direction parameter θ without an immediate statement of its admissible range; adding a short sentence clarifying θ ∈ [0, π] would improve readability.
  2. [Figure 3] Figure 3: The curvature plot lacks an explicit scale bar for the vertical axis; readers cannot directly compare the reported bound values to the plotted curve without additional measurement.
  3. [Theorem 2] The proof of local support in Theorem 2 relies on the compact support of the kernel but does not explicitly bound the size of the endpoint neighborhoods in terms of the smoothing scale ε; a one-line estimate would strengthen the claim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the directional mollification operator, its significance for applications in robotics and geometric modeling, and the recommendation for minor revision. No specific major comments appear in the provided report, so we have no point-by-point rebuttals to offer. We will address any minor editorial or clarification requests in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines a new directional mollification operator explicitly through a parametric family of kernels chosen to enforce vertex preservation by construction, local support, and C^∞ smoothness. All claimed properties (arbitrary closeness on compact sets, closed-form curvature bounds via explicit integrals, and the unified framework containing standard mollification as a special case) follow directly from the kernel definitions and standard mollification theory without any reduction to fitted inputs, self-citations, or renaming of prior results. The derivation is self-contained as an explicit construction with verifiable analytic properties for arbitrary polygonal chains.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of the directional mollification operator and its claimed properties, which are introduced without detailed construction in the abstract.

free parameters (1)
  • smoothing scale parameter
    Controls neighborhood size for approximation closeness and curvature; chosen or tuned per application.
axioms (1)
  • standard math Standard mollifier properties and convolution yield C^∞ approximations to locally integrable functions.
    Invoked to guarantee infinite differentiability and uniform approximation on compact sets.
invented entities (1)
  • directional mollification operator no independent evidence
    purpose: To achieve vertex-preserving C^∞ smoothing of polygonal chains with local support and explicit curvature bounds.
    Newly postulated operator whose properties are asserted in the abstract without external independent evidence.

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

  1. [1]

    Applied Mathematics and Computation 458, 128240

    On g1 and g2 hermite interpolation by spatial algebraic-trigonometric pythagorean hodo- graph curves with polynomial parametric speed. Applied Mathematics and Computation 458, 128240. URL: https://www.sciencedirect. com/science/article/pii/S0096300323004095, doi:https://doi.org/10.1016/j.amc.2023.128240. Bentamy, A., Guibault, F., Trépanier, J.Y.,

    work page doi:10.1016/j.amc.2023.128240 2023

  2. [2]

    Computer-Aided Design 153, 103408

    An analytical method for corner smoothing of five-axis linear paths using the conformal geometric algebra. Computer-Aided Design 153, 103408. URL: https://www.sciencedirect.com/science/article/pii/S0010448522001415, doi:https://doi.org/10.1016/j.cad.2022.103408. Cohen, E., Riesenfeld, R.F., Elber, G.,

    work page doi:10.1016/j.cad.2022.103408 2022

  3. [3]

    Computer Aided Geometric Design 4, 125–131

    Cutting corners always works. Computer Aided Geometric Design 4, 125–131. URL: https://www.sciencedirect.com/ science/article/pii/016783968790029X, doi:https://doi.org/10.1016/0167-8396(87)90029-X . topics in CAGD. Dyn, N., Hormann, K., Mancinelli, C.,

    work page doi:10.1016/0167-8396(87)90029-x

  4. [4]

    Computer Aided Geometric Design 94, 102083

    Non-uniform interpolatory subdivision schemes with improved smoothness. Computer Aided Geometric Design 94, 102083. URL: https://www.sciencedirect.com/science/article/pii/S016783962200019X, doi: https: //doi.org/10.1016/j.cagd.2022.102083. Evans, L.C.,

    work page doi:10.1016/j.cagd.2022.102083 2022

  5. [5]

    and Müller, Peter , title = "

    URL: https://doi.org/10.1111/j.1467-9876.2007.00592.x, doi: 10.1111/j.1467-9876.2007.00592.x, arXiv:https://academic.oup.com/jrsssc/article-pdf/56/5/571/50008302/jrsssc_56_5_571.pdf. Farin, G.E.,

    work page doi:10.1111/j.1467-9876.2007.00592.x 2007

  6. [6]

    Computer Aided Geometric Design 124, 102505

    A quaternion approach to g2 hermite interpolation by quintic spatial pythagorean-hodograph curves. Computer Aided Geometric Design 124, 102505. URL: https://www.sciencedirect.com/science/article/pii/S0167839625000949, doi: https: //doi.org/10.1016/j.cagd.2025.102505. Farouki, R.T., Gentili, G., Giannelli, C., Sestini, A., Stoppato, C.,

    work page doi:10.1016/j.cagd.2025.102505 2025

  7. [7]

    URL: https://arxiv.org/abs/2512.13183, arXiv:2512.13183

    Efficient generation of smooth paths with curvature guarantees by mollification. URL: https://arxiv.org/abs/2512.13183, arXiv:2512.13183. Hameed, R., Mustafa, G.,

    work page arXiv

  8. [8]

    Applied Mathematics and Computation 309, 289–302

    Family of a-point b-ary subdivision schemes with bell-shaped mask. Applied Mathematics and Computation 309, 289–302. URL: https://www.sciencedirect.com/science/article/pii/S0096300317302576, doi:https://doi.org/10.1016/ j.amc.2017.04.013. Hasan, M.S., Alam, M.N., Fayz-Al-Asad, M., Muhammad, N., Tunç, C.,

    work page 2017

  9. [9]

    Gradient-informed path smoothing for wheeled mobile robots, in: 2018 IEEE International Conference on Robotics and Automation (ICRA), IEEE. pp. 1–9. Hormann, K., Mancinelli, C.,

    work page 2018

  10. [10]

    Minimum snap trajectory generation and control for quadrotors, in: 2011 IEEE International Conference on Robotics and Automation (ICRA), IEEE. pp. 2520–2525. Ohrhallinger, S., Peethambaran, J., Parakkat, A.D., Dey, T.K., Muthuganapathy, R.,

    work page 2011

  11. [11]

    Robotics and Computer-Integrated Manufacturing 72, 102193

    An analytical method for decoupled local smoothing of linear paths in industrial robots. Robotics and Computer-Integrated Manufacturing 72, 102193. URL: https://www.sciencedirect.com/science/article/pii/ S0736584521000764, doi:https://doi.org/10.1016/j.rcim.2021.102193. Piegl, L., Tiller, W.,

    work page doi:10.1016/j.rcim.2021.102193 2021

  12. [12]

    International Journal of Automation Technology 15, 621–630

    Kinematic tool-path smoothing for 6-axis industrial machining robots. International Journal of Automation Technology 15, 621–630. doi: 10.20965/ijat.2021.p0621. Tan, J., Tong, G., Zhang, L., Xie, J.,

    work page doi:10.20965/ijat.2021.p0621 2021

  13. [13]

    Applied Mathematics and Computation 265, 819–825

    Four point interpolatory-corner cutting subdivision. Applied Mathematics and Computation 265, 819–825. URL: https://www.sciencedirect.com/science/article/pii/S0096300315007389, doi:https://doi.org/10.1016/ j.amc.2015.05.107. A. González-Calvin, J. F. Jiménez, and H. G. de Marina: Preprint submitted to Elsevier Page 18 of 18

    work page 2015