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arXiv · 2603.21831

Directional Mollification for Knot-Preserving $C^{\infty}$ Smoothing of Polygonal Chains with Explicit Curvature Bounds

Alfredo González-Calvin·Juan F. Jiménez·Héctor García de Marina
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Abstract

We introduce the \textit{directional mollification} operator, which acts on polygonal chains to produce $C^{\infty}$ curve approximants that are arbitrarily close to the original curve -- pointwise and uniformly on compact subsets -- while strictly interpolating its vertices. Unlike standard mollification, which fails to preserve vertices, this directional construction enables local, vertex-preserving smoothing; modifying a single segment alters the $C^{\infty}$ output only within an explicitly controllable neighborhood. The operator admits closed-form curvature bounds and provides analytic control over curve geometry. Furthermore, we develop a parametric family of smoothing operators that unifies conventional and directional mollification into a single framework. Combining computational simplicity with strict waypoint fidelity, this method is directly applicable to geometric modeling, robotics, computer graphics, and CNC machining.

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Alfredo González-Calvin, Juan F. Jiménez, Héctor García de Marina. 2026-03-23. Directional Mollification for Knot-Preserving $C^{\infty}$ Smoothing of Polygonal Chains with Explicit Curvature Bounds. https://arxiv.org/abs/2603.21831

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