提出方向性光滑方法，实现多段线的无限光滑且严格保持顶点。
Directional Mollification for Knot-Preserving $C^{\infty}$ Smoothing of Polygonal Chains with Explicit Curvature Bounds
- 基于方向性构造的光滑算子，可局部控制平滑区域。
- 生成曲线在紧集上一致逼近原曲线，且顶点完全保留。
- 提供显式曲率界，适合机器人路径规划等高精度场景。
我们提出一种方向性光滑算子，作用于多段线以生成与原始曲线在紧集上点态和一致逼近的 $C^{ u}$ 曲线近似，同时严格插值其顶点。与传统光滑方法不同，该方向性构造实现局部顶点保持平滑；仅修改一段时，$C^{ u}$ 输出的变化被明确控制在邻域内。该算子具有闭式曲率界，提供对曲线几何的解析控制。此外，我们构建了一个参数化光滑算子族，将经典与方向性光滑统一于同一框架。兼具计算简便与关键点保真性，该方法可直接应用于几何建模、机器人路径规划、计算机图形学及数控加工。
原文摘要 · Abstract (English)
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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