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.
From: Alfredo González Calvin [view email]
[v1]
Mon, 23 Mar 2026 11:18:43 UTC (425 KB)
[v2]
Mon, 13 Apr 2026 08:07:13 UTC (816 KB)
[v3]
Wed, 22 Jul 2026 15:44:15 UTC (850 KB)
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