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

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.