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

cs.ROMar 23, 2026

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

Authors: Alfredo González-Calvin, Juan F. Jiménez, Héctor García de Marina

Organizations: Dept. of Computer Architecture and Automation, Faculty of Physics, Complutense University of Madrid, Plaza de las Ciencias, 1, Madrid, 28040, Madrid, Spain · Dept. of Computer Engineering, Automation, and Robotics& Institute of Mathematics, University of Granada, Calle Ventanilla, Granada, 18001, Granada, Spain

Abstract

We introduce the 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.

Figures & tables

Figure 1: Example of mollification of a polygonal chain

f:\mathbb{R}\to\mathbb{R}^{2}

, where the coordinates in

\mathbb{R}^{2}

are expressed as

(u,v)

. The green curve shows the (conventionally) mollified function

F_{0.4}

as in Theorem 2 with the mollifier presented in Example 1 . The number

0.4

, stands for a parameter

\varepsilon=0.4

in which the mollification depends. The original (black) curve is an illustrative case of a polygonal chain curve whose knots/vertices are shown as black dots. As can be seen, the mollified (green) curve is contained in the convex hull (blue filled polygon) of the image of the original curve as Theorem 2 states, but it does not intersect the vertices. This is a trade-off between smoothness, convergence, and the approximation provided by the conventional mollification with ( 1 ). Nevertheless, it will be shown in Section 5 that for this kind of curve, our directional mollification method will generate a smooth curve that inherits the convergence and smoothness properties of the conventional mollification while intersecting all the knots/vertices that define the original polygonal chain curve, while obtaining explicit curvature estimates. This is illustrated by the red plot, with the directional mollification

\widehat{F}_{0.4}

introduced and studied in this paper.

Property / Case	Conventional Mollification	Directional Mollification
Assumptions	$f$ locally integrable	$f$ locally integrable and directionally differentiable at all points with locally integrable directional derivative
Equation	$F_{\varepsilon}(x)=\int_{\mathbb{R}^{n}}f(s)\varphi_{\varepsilon}(x-s)ds$	$\widehat{F}_{\varepsilon}(x)=\int_{\mathbb{R}^{n}}(f(s)+f^{\circ}(s)(x-s))\varphi_{\varepsilon}(x-s)ds$
Regularity	$F_{\varepsilon}\in C^{\infty}$ , Theorem 1	If $\nabla f$ exists a.e., $\widehat{F}_{\varepsilon}\in C^{\infty}$ , Theorem 4
Convergence on compact subsets	$f$ continuous, Theorem 1	$f$ locally Lipschitz, Theorem 4
Affine $f$ and even mollifier	$F_{\varepsilon}=f$ for all $\varepsilon>0$ , Theorem 2	$\widehat{F}_{\varepsilon}=f$ for all $\varepsilon>0$ , Theorem 4
Convex $f$	$F_{\varepsilon}$ is convex and $f\leq F_{\varepsilon}$ for all $\varepsilon>0$ , Theorem 2	$\widehat{F}_{\varepsilon}$ is not necessarily convex for all $\varepsilon>0$ , but $\widehat{F}_{\varepsilon}\leq f$ for all $\varepsilon>0$ , Theorem 4

Table 1: A summary of the primary distinctions between the conventional and directional mollification approaches. It is always assumed that

\varphi\in\mathscr{D}_{M}(\mathbb{R}^{n})

Figure 2: Representation of the conventional and directional mollification of the strictly increasing function $f(x)=x\ind_{(-\infty,0)}(x)+30x\ind_{[0,\infty)}(x)$ (left picture) and the function $t\in[0,2\pi]\mapsto(2+\cos(2t))(\cos(t),\sin(t))\in\mathbb{R}^{2}$ (right picture, where the coordinates of $\mathbb{R}^{2}$ are shown as $(u,v)$ ), where in both cases the used mollifier $\varphi$ is as in ( 1 ). For the latter, the convolution is carried out by extending its domain to $\mathbb{R}$ . In the left picture, the black plot represents the original function $f$ , while the green one is its conventional mollification, and the red one is its directional mollification, both with $\varepsilon=0.5$ . The right picture represents the original function $f$ as a black plot, while the blue and red plots the directional mollifications $\widehat{F}_{\varepsilon}$ for $\varepsilon=1.25$ and $\varepsilon=5$ , respectively.

Figure 3: Representation of the conventional and directional mollification approaches for a

p=6

polygonal chain in

\mathbb{R}^{2}

defined as in ( 2 ), where the coordinates of

\mathbb{R}^{2}

are expressed as

(u,v)

. The top left picture represents in black the original polygonal chain curve

f

, in dashed green its conventional mollification

F_{0.4}

, in red its directional mollification

\widehat{F}_{0.4}

, and as a blue transparent patch the convex hull of the vertices. In all cases, the mollifier used is the one presented in ( 1 ). The top right and bottom left pictures represent, respectively, the first and second components of the polygonal chain, as well as its conventional and directional mollifications with the same color codes as in the top-left picture. The solid red regions represent the sets

V_{r}

of Theorem 6 for

r\in\{1,\dots,p\}

. The bottom right picture represents the errors (computed using the Euclidean norm) between the original function and the conventional mollification in dashed green, and the directional mollification in red. The solid red regions represent, as in the top-right and bottom-left pictures, the sets

V_{r}

Figure 4: Representation of a

p=7

polygonal chain in

\mathbb{R}^{3}

, and the curvature and curvature bounds of the conventional and directional mollifications. The coordinates in

\mathbb{R}^{3}

are expressed as

(u,v,w)

. The left picture represents in black the polygonal chain curve

f

confined in a unit volume cube in

\mathbb{R}^{3}

, shown in blue. It also represents its conventional

F_{\varepsilon}

and directional mollifications

\widehat{F}_{\varepsilon}

in dashed green and solid red, respectively. In all mollifications, the used mollifier is the one presented in ( 1 ), with

\varepsilon=0.4

. Right picture represents in green the curvature of the conventional mollified curve

F_{\varepsilon}

, and as a dashed green plot, the curvature upper bounds

\kappa_{F_{\varepsilon}}

for each pair of segment as follows. In the set

[0,1.5]

the curvature bound is computed using ( 11 ) for the polygonal chain consisting of the points

P_{0},P_{1}

and

P_{2}

. In the set

[k+0.5,k+1.5]

the curvature bound is computed using ( 11 ) for the polygonal chain consisting of the points

P_{k},P_{k+1}

and

P_{k+2}

for

k\in\{1,\dots,p-1\}

. The solid and red dashed plots represent, respectively, the curvature of

\widehat{F}_{\varepsilon}

, and its curvature upper bounds

\kappa_{\widehat{F}_{\varepsilon}}

computed in the same manner as for the conventional mollification, but using the formula ( 7 ) obtained in Proposition 9 .

Figure 5: Illustrative comparison of a family of curves using ( 15 ), where the mollifier used is the one presented in Example 1 with

\varepsilon=0.4

and different approaches of polynomial splines. The left picture represents in black a

p=6

polygonal chain curve in

\mathbb{R}^{2}

whose vertices are represented as black dots, and the components of

\mathbb{R}^{2}

(u,v)

. It also shows in blue a cubic spline, in red a B-spline, and in green a Quintic Hermite polynomial spline. All of them are created such that they intersect the knots/vertices. The center picture also displays the original function as a black plot, while the red and blue curves show the conventional and directional mollifications of

f

, respectively. The dashed green lines represent

G_{\varepsilon}^{\gamma}

as in ( 15 ), for the values

\gamma\in\{-1,-0.5,0,0.5,1,1.5,2\}

. The right picture represents the Euclidean norms between the original function and the different approaches. That is, it represents the function

t\mapsto\|h(t)-f(t)\|_{2}

, where,

h

is either, the cubic spline (blue dashed plot), B-spline (red dashed plot), quintic Hermite spline (green dashed plot), the conventional mollification (red solid plot) or the directional mollification (black plot) shown in the left and center pictures. The red solid regions represent the sets

V_{r}

of Theorem 6 for

r\in\{1,\dots,p\}

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Centre for Visual Computing and Intelligent Systems University of Bradford United Kingdom · School of Individualized Study Rochester Institute of Technology United States

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1McGill University · 2National University of Mongolia · Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences

ReLU Neural Networks Neural Network Approximation Theory Iterative Refinement

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