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

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.