A Geometric Gaussian Mixture Representation of Plane Curves

# A Geometric Gaussian Mixture Representation of Plane Curves
Source: [https://arxiv.org/html/2606.06505](https://arxiv.org/html/2606.06505)
###### Abstract

We introduce a user defined probabilistic polygonal representation for plane curves\. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation\. Each segment is equipped with a user defined uncertainty parameter \(standard deviation\) in the normal direction\. This yields a collection of thin probabilistic geometric primitives that retain the local tangent, normal, arc length of the underlying plane curve while extending it beyond the idealized deterministic one dimensional formulation\.

For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment\. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty\. Combining the segment wise components with appropriate weights yields a Gaussian Mixture Model representation of the user defined probabilistic polygonal representation of the plane curve\.

The proposed framework provides an analytically tractable probabilistic model that preserves local position, orientation, length scale, and uncertainty in the normal direction\. It applies to smooth, closed, open, non regular, and self intersecting plane curves, allows adaptive discretization and varying uncertainty in the normal direction, and as a result supports uncertainty aware geometric modeling\. Experiments on a collection of canonical plane curves show that the resulting Gaussian Mixture Model capture local tangent, local normal, and local arc length; resulting in the global shape of the underlying curves to be truthfully captured as well\. The representation is particularly relevant for applications in uncertainty aware CAD and digital twins, probabilistic obstacle modeling in robotics, and probabilistic trajectory planning\.

## 1Introduction

Plane curves while the simplest geometric objects in modern geometry\[[10](https://arxiv.org/html/2606.06505#bib.bib14)\]are fundamental in a wide range of applications, including computer vision\[[14](https://arxiv.org/html/2606.06505#bib.bib15)\], robotics\[[12](https://arxiv.org/html/2606.06505#bib.bib16)\], measurement systems\[[5](https://arxiv.org/html/2606.06505#bib.bib20),[1](https://arxiv.org/html/2606.06505#bib.bib6)\], shape analysis\[[2](https://arxiv.org/html/2606.06505#bib.bib17)\], motion and path planning\[[9](https://arxiv.org/html/2606.06505#bib.bib21)\], and computer aided design\[[13](https://arxiv.org/html/2606.06505#bib.bib18),[4](https://arxiv.org/html/2606.06505#bib.bib19)\]\. In many such settings, plane curves are used to describe object boundaries, feature contours, nominal paths, or measurement systems calibration curves\. Classical representations such as parameterized form, polygonal approximation\[[16](https://arxiv.org/html/2606.06505#bib.bib91)\], polynomials, and splines\[[4](https://arxiv.org/html/2606.06505#bib.bib19),[13](https://arxiv.org/html/2606.06505#bib.bib18)\]are well suited for describing plane curves and for supporting numerical computation\. However, they are typically deterministic: they specify where a curve lies, but do not directly encode in the normal direction of the curve\.

At the same time, probabilistic representations play a central role in estimation, inference, and uncertainty aware modeling\.*Gaussian Mixture Model*s provide a flexible and analytically tractable class of parametric*Probability Density Function*s for representing uncertainty\. Without knowing anything about the problem*Gaussian Mixture Model*s are are usually fitted to point samples \(data\) and therefore do not, by themselves, exploit the prior knowledge that the data lie on a curve\. As a result, there remains a gap between deterministic curve representations, which retain geometric meaning, and probabilistic models, which support uncertainty aware modeling that involves inducing uncertainty to the deterministic curve\.

In this work, we propose a structured way to bridge this gap for a special case\. Starting from a plane curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal representation of the curve\. In contrast to a purely deterministic polygonal chain, each segment is assigned a uncertainty parameter in the normal direction of the segment that models uncertainty, or tolerance\. This yields a user defined probabilistic polygonal representation of the curve: a collection of thin geometric primitives that preserve local position and orientation while also carrying an explicit notion of uncertainty in the tangent and in the normal direction of the segment\.

For each segment, we define a*Random Variable*that distributes the probability measure uniformly along the segment tangent direction and normally in the segment normal direction\. This construction leads, through exact first and second central moment calculations, to Gaussian*Probability Density Function*whose mean lies at the segment midpoint and whose covariance captures both tangential uncertainty and normal uncertainty\. By mixing these Gaussian components appropriate weights, we obtain a*Gaussian Mixture Model*representation of the entire curve\. The resulting model is directly derived from user defined probabilistic polygonal representation parameters rather than obtained by iterative fitting to sampled point sets\.

The proposed construction preserves local and global shape in a probabilistic and analytically tractable form\. In particular, it holds segment position, segment tangent and normal direction, segment arc length scale, and uncertainty in the normal direction of the segment through the*Gaussian Mixture Model*representation\. The construction applies to smooth, closed, open, non regular, periodic, and self intersecting rectifiable plane curves, requires only the evaluation of curve points at partition nodes, and naturally supports adaptive discretization and varying uncertainty in the normal direction of the segments\. In this way, deterministic plane curves get first transformed into a user defined probabilistic polygonal representation and then transformed into a*Gaussian Mixture Model*\.

This representation is relevant in several application domains\. In uncertainty aware CAD and digital twins, user defined uncertainty in the normal direction of the segment can represent manufacturing tolerances or wear\. In robotics and autonomous systems, probabilistic polygonal approximations of curves provide representations of boundaries of obstacle contours with safety margins or sensor uncertainty\. In trajectory modeling and statistical shape analysis, the induced*Gaussian Mixture Model*s furnish compact priors that preserve nominal geometry while allowing controlled uncertainty normal to the boundary of the object\.

Our main contributions are as follows:

- •We introduce a user defined probabilistic polygonal representation of plane curves in which each polygonal segment is equipped with a uncertainty in the normal direction of the segment\.
- •We replace the*Probability Density Function*of the user defined probabilistic segment with a Gaussian and derive closed form expressions for the mean and covariance matrix of the Gaussian in terms of the user defined probabilistic segment parameters\.
- •We construct a*Gaussian Mixture Model*representation of the user defined probabilistic polygonal representation of the curve whose components parameters are derived again in closed form\.
- •We demonstrate that the framework applies to a broad range of canonical plane curves and can capture their local and the global shape, including smooth, closed, open, non regular, singular, and self intersecting ones, and supports adaptive discretization as well as varying uncertainty in the normal direction of the segments\.

The remainder of the paper is organized as follows\. In Section[3](https://arxiv.org/html/2606.06505#S3), we formally introduce the user defined probabilistic polygonal representation of a plane curve\. We then derive the per segment Gaussian*Probability Density Function*and the induced*Gaussian Mixture Model*construction\. Finally in[Section˜4](https://arxiv.org/html/2606.06505#S4)we illustrate the framework on representative examples and discuss modeling choices, limitations, and possible extensions\.

### 1\.1Technical Applications

Beyond the theoretical formulation, the proposed construction admits several concrete application scenarios\. In computer aided design and digital twinning, edges and intersection curves of CAD models can be mapped deterministically to*Gaussian Mixture Model*s, yielding a differentiable “Gaussian splat” representation of ideal geometry\. User defined uncertainties are encoded directly in the uncertainty parametersτi\\tau\_\{i\}, enabling uncertainty aware rendering, tolerance analysis, and geometric reasoning in a form that is compatible with recent Gaussian splatting based scene representations\[[7](https://arxiv.org/html/2606.06505#bib.bib261)\]\.

In robotics and autonomous driving, free space and obstacle boundaries extracted from LiDAR or range data are typically modeled via occupancy grids or distance fields\[[11](https://arxiv.org/html/2606.06505#bib.bib264)\]\. Our framework offers an alternative boundary centric view: local obstacle contours are represented as probabilistic tubular neighborhoods around user defined probabilistic polygonal representation of curves, where sensor noise, safety margins, or learned epistemic uncertainty are encoded intrinsically throughτi\\tau\_\{i\}\. This connects naturally to recent work that uses*Gaussian Mixture Model*s as compact, expressive occupancy models\[[6](https://arxiv.org/html/2606.06505#bib.bib262),[18](https://arxiv.org/html/2606.06505#bib.bib263)\], but differs in that the mixture components are obtained in closed form from geometric primitives rather than via iterative fitting\. More broadly, the construction provides a generic way to endow deterministic plane curves with a tunable transverse uncertainty field while retaining an analytically tractable*Gaussian Mixture Model*structure suitable for inference, optimization, and control\.

## 2Related Work

We focus here on work related to constructing*Gaussian Mixture Model*representations of plane curves under uncertainty\. Although some existing methods are not formulated directly for parametrized curves, they can often be applied after first converting the curve into a point cloud by sampling its parameter domain\. In this way, a parametrized curve can be reduced to a finite collection of points, after which point cloud based*Gaussian Mixture Model*methods such as\[[3](https://arxiv.org/html/2606.06505#bib.bib3)\],\[[15](https://arxiv.org/html/2606.06505#bib.bib4)\], and related approaches become applicable\. These methods typically assume noisy point data and construct a*Gaussian Mixture Model*representation using EM or its suitable modifications\.

A related but more geometry aware approach is presented in\[[8](https://arxiv.org/html/2606.06505#bib.bib2)\], where the underlying manifold is given as a triangular mesh\. There, the*Gaussian Mixture Model*construction is informed by the mesh structure, and the EM procedure is modified accordingly\. In this sense, the method combines data driven fitting with an explicit geometric representation of the surface\.

## 3Problem Formulation and the Gaussian Mixture Model Construction

For the convenience of the reader, we begin by explicitly stating several standard notions from differential geometry of curves in order to keep the presentation self contained\. Readers already familiar with this material may wish to proceed more quickly through the following definitions and remarks and focus on the user defined probabilistic polygonal representation and the*Gaussian Mixture Model*construction that it induces\.

###### Definition 3\.1\(Parameter Interval\)\.

A parameter interval is a compact interval

I=\[a,b\]⊂ℝ,a,b∈ℝ∧a<b\.I=\[a,b\]\\subset\\mathbb\{R\},\\qquad a,b\\in\\mathbb\{R\}\\land a<b\.\(1\)

###### Definition 3\.2\(Plane Curve\)\.

LetI⊂ℝI\\subset\\mathbb\{R\}be a parameter interval\. A plane curve is a map

α:I→ℝ2\.\\alpha:I\\to\\mathbb\{R\}^\{2\}\.\(2\)

###### Definition 3\.3\(Partition and Induced Polygonal Approximation\)\.

LetI⊂ℝI\\subset\\mathbb\{R\}be a parameter interval and letα:I→ℝ2\\alpha:I\\to\\mathbb\{R\}^\{2\}be a plane curve\. A partition ofIIis a finite ordered set

t0<t1<⋯<tN,ti∈I,t\_\{0\}<t\_\{1\}<\\cdots<t\_\{N\},\\qquad t\_\{i\}\\in I,\(3\)such that

I=⋃i=1N\[ti−1,ti\]\.I=\\bigcup\_\{i=1\}^\{N\}\[t\_\{i\-1\},t\_\{i\}\]\.\(4\)The corresponding vertices on the curve are defined by

vi≔α​\(ti\),i=0,…,N,v\_\{i\}\\coloneq\\alpha\(t\_\{i\}\),\\qquad i=0,\\dots,N,\(5\)and the induced polygonal segments are

Si≔\(vi−1,vi\),i=1,…,N\.S\_\{i\}\\coloneq\(v\_\{i\-1\},v\_\{i\}\),\\qquad i=1,\\dots,N\.\(6\)

###### Definition 3\.4\(Uncertainty Parameter in the Normal Direction\)\.

For each user defined segmentSiS\_\{i\}, an uncertainty parameter in the normal direction of the segment is a scalar

τi\>0,i=1,…,N\.\\tau\_\{i\}\>0,\\qquad i=1,\\dots,N\.\(7\)

###### Definition 3\.5\(User Defined Probabilistic Polygonal Representation of a Plane Curve\)\.

Letα:I→ℝ2\\alpha\\colon I\\to\\mathbb\{R\}^\{2\}be a rectifiable plane curve, together with a partition induced segments\{Si\}i=1N\\\{S\_\{i\}\\\}\_\{i=1\}^\{N\}and associated uncertainty in the normal direction of the segment parameters\{τi\}i=1N\\\{\\tau\_\{i\}\\\}\_\{i=1\}^\{N\}\. The collection

𝒫=\{\(Si,τi\)\}i=1N\\mathcal\{P\}=\\\{\(S\_\{i\},\\tau\_\{i\}\)\\\}\_\{i=1\}^\{N\}\(8\)is called a user defined probabilistic polygonal representation ofα\\alpha\.

###### Definition 3\.6\(Per Segment Random Variable\)\.

For a segmentSi=\(vi−1,vi\)S\_\{i\}=\(v\_\{i\-1\},v\_\{i\}\), define

ℓi:=‖vi−vi−1‖,ei:=vi−vi−1ℓi\(ℓi\>0\),\\ell\_\{i\}:=\\\|v\_\{i\}\-v\_\{i\-1\}\\\|,\\qquad e\_\{i\}:=\\frac\{v\_\{i\}\-v\_\{i\-1\}\}\{\\ell\_\{i\}\}\\quad\(\\ell\_\{i\}\>0\),\(9\)and let

ni:=R​ei,R=\(0−110\)\.n\_\{i\}:=Re\_\{i\},\\qquad R=\\begin\{pmatrix\}0&\-1\\\\ 1&\\phantom\{\-\}0\\end\{pmatrix\}\.\(10\)Forτi≥0\\tau\_\{i\}\\geq 0, the*Random Variable*associated withSiS\_\{i\}is

Xi:=vi−1\+ℓi​Ui​ei\+τi​Zi​ni,X\_\{i\}:=v\_\{i\-1\}\+\\ell\_\{i\}U\_\{i\}e\_\{i\}\+\\tau\_\{i\}Z\_\{i\}n\_\{i\},\(11\)where

Ui∼𝒰​\(ui\|0,1\),Zi∼𝒩​\(zi\|0,1\),Ui⟂Zi\.U\_\{i\}\\sim\\mathcal\{U\}\(u\_\{i\}\|0,1\),\\qquad Z\_\{i\}\\sim\\mathcal\{N\}\(z\_\{i\}\|0,1\),\\qquad U\_\{i\}\\perp Z\_\{i\}\.\(12\)

###### Proposition 3\.1\(Expectation and Variance of a User Defined Probabilistic Segment\)\.

LetXiX\_\{i\}be defined as above\. Then its expectation is

𝔼​\(Xi\)=vi−1\+vi2,\\mathbb\{E\}\(X\_\{i\}\)=\\frac\{v\_\{i\-1\}\+v\_\{i\}\}\{2\},\(13\)and its Variance is

𝕍​\(Xi\)=ℓi212​ei​ei⊤\+τi2​ni​ni⊤\.\\mathbb\{V\}\(X\_\{i\}\)=\\frac\{\\ell\_\{i\}^\{2\}\}\{12\}e\_\{i\}e\_\{i\}^\{\\top\}\+\\tau\_\{i\}^\{2\}n\_\{i\}n\_\{i\}^\{\\top\}\.\(14\)

###### Proof\.

By independence ofUiU\_\{i\}andZiZ\_\{i\}, and using

𝔼​\(Ui\)=12,𝕍​\(Ui\)=112,𝔼​\(Zi\)=0,𝕍​\(Zi\)=1,\\mathbb\{E\}\(U\_\{i\}\)=\\frac\{1\}\{2\},\\qquad\\mathbb\{V\}\(U\_\{i\}\)=\\frac\{1\}\{12\},\\qquad\\mathbb\{E\}\(Z\_\{i\}\)=0,\\qquad\\mathbb\{V\}\(Z\_\{i\}\)=1,\(15\)we obtain

𝔼​\(Xi\)=vi−1\+𝔼​\(Ui\)​\(vi−vi−1\)\+τi​𝔼​\(Zi\)​ni=vi−1\+12​\(vi−vi−1\)=vi−1\+vi2\.\\mathbb\{E\}\(X\_\{i\}\)=v\_\{i\-1\}\+\\mathbb\{E\}\(U\_\{i\}\)\(v\_\{i\}\-v\_\{i\-1\}\)\+\\tau\_\{i\}\\mathbb\{E\}\(Z\_\{i\}\)\\,n\_\{i\}=v\_\{i\-1\}\+\\frac\{1\}\{2\}\(v\_\{i\}\-v\_\{i\-1\}\)=\\frac\{v\_\{i\-1\}\+v\_\{i\}\}\{2\}\.\(16\)Moreover,

𝕍​\(Xi\)=𝕍​\(Ui​di\)\+𝕍​\(τi​Zi​ni\),\\mathbb\{V\}\(X\_\{i\}\)=\\mathbb\{V\}\(U\_\{i\}d\_\{i\}\)\+\\mathbb\{V\}\(\\tau\_\{i\}Z\_\{i\}n\_\{i\}\),\(17\)since the two terms are independent\. Therefore,

𝕍​\(Ui​\(vi−vi−1\)\)=Var​\(Ui\)​\(vi−vi−1\)​\(vi−vi−1\)⊤=112​\(vi−vi−1\)​\(vi−vi−1\)⊤,\\mathbb\{V\}\(U\_\{i\}\(v\_\{i\}\-v\_\{i\-1\}\)\)=\\mathrm\{Var\}\(U\_\{i\}\)\\,\(v\_\{i\}\-v\_\{i\-1\}\)\(v\_\{i\}\-v\_\{i\-1\}\)^\{\\top\}=\\frac\{1\}\{12\}\(v\_\{i\}\-v\_\{i\-1\}\)\(v\_\{i\}\-v\_\{i\-1\}\)^\{\\top\},\(18\)and

𝕍​\(τi​Zi​ni\)=τi2​𝕍​\(Zi\)​ni​ni⊤=τi2​ni​ni⊤\.\\mathbb\{V\}\(\\tau\_\{i\}Z\_\{i\}n\_\{i\}\)=\\tau\_\{i\}^\{2\}\\mathbb\{V\}\(Z\_\{i\}\)\\,n\_\{i\}n\_\{i\}^\{\\top\}=\\tau\_\{i\}^\{2\}n\_\{i\}n\_\{i\}^\{\\top\}\.\(19\)Hence,

𝕍​\(Xi\)=112​di​di⊤\+τi2​ni​ni⊤\.\\mathbb\{V\}\(X\_\{i\}\)=\\frac\{1\}\{12\}d\_\{i\}d\_\{i\}^\{\\top\}\+\\tau\_\{i\}^\{2\}n\_\{i\}n\_\{i\}^\{\\top\}\.\(20\)Ifℓi\>0\\ell\_\{i\}\>0, the identitydi=ℓi​eid\_\{i\}=\\ell\_\{i\}e\_\{i\}yields

𝕍​\(Xi\)=ℓi212​ei​ei⊤\+τi2​ni​ni⊤\.\\mathbb\{V\}\(X\_\{i\}\)=\\frac\{\\ell\_\{i\}^\{2\}\}\{12\}e\_\{i\}e\_\{i\}^\{\\top\}\+\\tau\_\{i\}^\{2\}n\_\{i\}n\_\{i\}^\{\\top\}\.\(21\)∎

###### Definition 3\.7\(Gaussian Mixture Model induced by a User Defined Probabilistic Polygonal Representation of a Plane Curve\)\.

Let𝒫=\{\(Si,τi\)\}i=1N\\mathcal\{P\}=\\\{\(S\_\{i\},\\tau\_\{i\}\)\\\}\_\{i=1\}^\{N\}be a probabilistic polygonal representation of a rectifiable plane curveα:I→ℝ2\\alpha:I\\to\\mathbb\{R\}^\{2\}\. The associated*Gaussian Mixture Model*is by construction:

𝒢​\(x\|g\)≔∑i=1Nπi​𝒩​\(x\|mi,Σi\),\\mathcal\{G\}\(x\|g\)\\coloneq\\sum\_\{i=1\}^\{N\}\\pi\_\{i\}\\mathcal\{N\}\(x\|m\_\{i\},\\Sigma\_\{i\}\),\(23\)where

mi=vi−1\+vi2,Σi=112​di​di⊤\+τi2​ni​ni⊤,πi=ℓi∑j=1Nℓj\.m\_\{i\}=\\frac\{v\_\{i\-1\}\+v\_\{i\}\}\{2\},\\qquad\\Sigma\_\{i\}=\\frac\{1\}\{12\}d\_\{i\}d\_\{i\}^\{\\top\}\+\\tau\_\{i\}^\{2\}n\_\{i\}n\_\{i\}^\{\\top\},\\qquad\\pi\_\{i\}=\\frac\{\\ell\_\{i\}\}\{\\sum\_\{j=1\}^\{N\}\\ell\_\{j\}\}\.\(24\)

## 4Examples and Discussion

In this section we discuss how this representation behaves on canonical curves, and how it can be interpreted in application settings\.

### 4\.1Illustrative Canonical Curves

The examples collected in[Appendix˜A](https://arxiv.org/html/2606.06505#A1)\(circle, ellipse, parabola, logarithmic spiral, cusp curve, cycloid, astroid, cardioid, lemniscate, rose curves, and various polygonal shapes\) demonstrate that the construction applies uniformly across several overlapping cases:

- •Smooth regular curves \(circle, ellipse, parabola, measurement curve, mirrored S\-curve, logarithmic spiral\): the the meansmim\_\{i\}form a standard polygonal approximation ofα\\alpha, and the covariance matrices rotate smoothly, with the dominant eigenvector following the tangent\. As the maximal segment lengthmaxi⁡ℓi\\max\_\{i\}\\ell\_\{i\}decreases, the resulting*Probability Density Function*concentrates in a thin tubular neighborhood around the curve\.
- •Curves with finite number of singularities \(semicubical cusp, astroid, cardioid\) that away from singular points the behavior is as above\. Near cusps, the tangent direction is ill\-defined at the limit, and the local mixture of neighboring components naturally reflects this ambiguity\. No special treatment is required in the construction beyond ensuring that the partition includes the singular points as vertices\.
- •Self intersecting and disconnected curves \(lemniscate, rose curves, disconnected semicircles, Yin–Yang\-type curves\) that the model does not assume global injectivity of the parametrization\. Different branches or disconnected components correspond simply to disjoint groups of segments\. At self\-intersections, the local*Probability Density Function*is a superposition of Gaussian components associated with distinct branches, each carrying its own tangent and normal directions\.

These examples illustrate that the representation remains meaningful for regular, non regular, self intersecting, and topologically disconnected curves, as long as the underlying partition is chosen appropriately\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x1.png)Figure 1:Simple gallery of fundamental curve types \(line segment, circle, ellipse, parabola, a bijective function, a mirrored tanh\-based curve, cardioid, superellipse, and a square\)\. For each curve, we show only the final*Gaussian Mixture Model**Probability Density Function*heatmap\. Per curve visual convergence and split views are provided in Appendix[A](https://arxiv.org/html/2606.06505#A1)\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x2.png)Figure 2:Complex gallery of intricate curve types \(Lissajous, lemniscate, rose curve with55petals, Yin\-Yang curve, downward logarithmic spiral, downward Archimedean spiral, astroid, cycloid, and the semicubical cusp\)\. For each curve, we show only the final*Gaussian Mixture Model**Probability Density Function*heatmap\. Per curve visual convergence and split views are provided in Appendix[A](https://arxiv.org/html/2606.06505#A1)\.Figure[1](https://arxiv.org/html/2606.06505#S4.F1)shows a simple gallery of fundamental curves, while Figure[2](https://arxiv.org/html/2606.06505#S4.F2)illustrates more intricate examples\. Detailed convergence behavior and split views for each curve are collected in Appendix[A](https://arxiv.org/html/2606.06505#A1)\.

### 4\.2Application Oriented Interpretations

#### 4\.2\.1Uncertainty Aware CAD and Digital Twins

Edges and feature curves extracted from CAD models can be mapped deterministically to*Gaussian Mixture Model*s, yielding a continuous probabilistic representation of nominal geometry\. The curveα\\alphaencodes the ideal design, while the thickness parametersτi\\tau\_\{i\}encode manufacturing tolerances, wear, or modeling uncertainty along different parts of the boundary\. The resulting Gaussian components can be interpreted as anisotropic “splat” primitives attached to edges, providing an analytically tractable alternative to mesh based or voxel based uncertainty models and aligning conceptually with recent Gaussian splatting representations in graphics and vision\[[7](https://arxiv.org/html/2606.06505#bib.bib261)\]\. In this setting, manually prescribed thickness fields allow engineers to encode tolerance classes directly into the probabilistic geometry\.

#### 4\.2\.2Probabilistic Obstacle Boundaries in Robotics

In mobile robotics and autonomous driving, environment representations are often grid based or voxel based, encoding per cell occupancy probabilities\[[11](https://arxiv.org/html/2606.06505#bib.bib264),[17](https://arxiv.org/html/2606.06505#bib.bib265)\]\. Our curve based construction offers a complementary boundary centric view: local obstacle contours extracted from LiDAR or depth data are represented as probabilistic polygonal curves\. Each segment carries a thicknessτi\\tau\_\{i\}that can encode sensor noise in the normal direction, safety margins, or learned epistemic uncertainty\. The induced*Gaussian Mixture Model*then serves as a continuous, differentiable approximation of an object avoidance map, enabling collision probability queries, risk aware trajectory optimization, and closed form propagation of uncertainty through linearized motion models\. Related work shows that*Gaussian Mixture Model*based occupancy models can achieve high information density and efficient communication\[[6](https://arxiv.org/html/2606.06505#bib.bib262),[18](https://arxiv.org/html/2606.06505#bib.bib263)\]\. In contrast, our approach obtains mixture parameters in closed form from geometric primitives, so the construction can be implemented as a lightweight and reliable subroutine even on embedded platforms\.

#### 4\.2\.3Trajectory Templates and Statistical Shape Models\.

Nominal motion patterns or shape templates can be stored as plane curves and converted to*Gaussian Mixture Model*s using the proposed construction\. This yields compact, interpretable priors for Bayesian filtering, prediction, or registration tasks: the component means encode the template geometry, whileτi\\tau\_\{i\}captures allowable deviations orthogonal to the nominal path or boundary\. In shape analysis, collections of annotated curves can be converted to families of*Gaussian Mixture Model*s, opening the door to mixture based statistical shape models and distance measures that respect both geometric structure and spatial uncertainty\.

### 4\.3Limitations and Open Problems

Several important issues remain open \(look at[Remark˜3\.10](https://arxiv.org/html/2606.06505#S3.Thmremark10)and[Remark˜3\.11](https://arxiv.org/html/2606.06505#S3.Thmremark11)\) in the present theoretical development\. will require further work, together with additional tools from functional analysis and geometry\. The purpose of the present paper is therefore more modest: to provide a clear and tractable starting point, namely a direct mapping from deterministic plane curves with user defined probabilistic polygonal representation of the plane curves to a*Gaussian Mixture Model*representation\.

#### 4\.3\.1Convergence and Consistency

The present construction is intended to be asymptotically well behaved under refinement of the underlying discretization\. In particular, under suitable regularity assumptions on the curve, sufficiently fine partitions, and appropriate control of the thickness field, one expects the polygonal approximation to converge to the underlying curve, the covariance matrices to remain aligned with the local tangent and normal directions, and the induced*Gaussian Mixture Model*to converge, in a suitable weak sense, to a probabilistic tubular representation of the curve\.

However, turning these informal consistency statements into rigorous mathematical results is nontrivial\. A satisfactory analysis would require precise assumptions, a careful choice of convergence notion, and quantitative control of approximation error, for example in terms of geometric error, probability metrics, or divergence based criteria\. It would also be necessary to understand how these questions interact with[Remark˜3\.11](https://arxiv.org/html/2606.06505#S3.Thmremark11),[Remark˜3\.10](https://arxiv.org/html/2606.06505#S3.Thmremark10)singularities, and self intersections\.

For these reasons, a full treatment of convergence rates, consistency, and approximation guarantees lies beyond the scope of the present paper\. Since the more immediate challenges are of an applied and computational nature, we leave a rigorous asymptotic analysis to future work\.

#### 4\.3\.2Extensions Beyond Plane Curves

Although the present work focuses on plane curves, the same ideas extend naturally to space curves and to surface inℝ3\\mathbb\{R\}^\{3\}\. In those settings, Gaussian components would encode tangent directions together with uncertainty in the corresponding normal direction\(s\), suggesting probabilistic representations of higher dimensional geometric objects\. Developing these extensions in a mathematically satisfactory and computationally effective way will be done in future works\.

#### 4\.3\.3Need for a New Calculus

The present construction replaces a deterministic curve by a*Gaussian Mixture Model*representation in the ambient space\. While this yields an analytically tractable probabilistic model, it also changes the nature of the object under study: instead of working directly with a smooth parametrized curve or manifold, one works with a finite mixture of Gaussian components\. As a consequence, many classical tools from differential geometry are no longer directly available in their standard form\. In particular, notions such as tangent bundles, connections, covariant derivatives, and Christoffel symbols are defined for smooth manifolds and tensor fields, not for*Gaussian Mixture Model*representations\.

This does not mean that the geometric information is lost; rather, it is encoded in a different form through component means, covariance matrices, and mixture weights\. Developing a mathematically satisfactory framework that plays, for*Gaussian Mixture Model*s, a role analogous to differential calculus on smooth manifolds therefore appears to be an important open problem\. Such a framework would ideally permit geometric notions such as curvature, transport, and compatibility with local orientations and uncertainty to be expressed directly at the level of the*Gaussian Mixture Model*representation\. We leave the development of such a calculus to future work\.

## 5Conclusion

We have developed mathematically explicit formulas to represent user defined probabilistic polygonal of a plane curve by*Gaussian Mixture Model*s\. Beginning with a reference user defined probabilistic segment that has uniform uncertainty in the tangent direction and Gaussian uncertainty in the normal direction, we derived its central moment matched Gaussian equivalent in closed form and since affine transformation of a Gaussian*Random Variable*is a Gaussian*Random Variable*with arithmetically calculable mean and covariance matrix; the construction is generalizable for arbitrary user defined probabilistic segment\. Consequently, each user defined probabilistic segment can be represented by a Gaussian component, yielding a*Gaussian Mixture Model*representation of the user defined user defined probabilistic polygonal representation of the curve\.

In contrast to the classical parametrized formulation of plane curves, the resulting*Gaussian Mixture Model*incorporates uncertainty at the level of the representation itself\. The resulting construction therefore provides a probabilistic description in which both the underlying curve and the user defined uncertainties are encoded\. This makes the framework a natural candidate for applications in which geometric objects are observed, estimated, or manipulated under uncertainty\.

The examples presented in the appendix indicate that the proposed construction preserves the shape of the original curve both locally and globally\. In particular, ellipses that are the representative of the covariance matrices and*Probability Density Function*visualizations show that the constructed*Gaussian Mixture Model*preserve the directional structure and the probabilistic nature of the user defined probabilistic polygonal representation of the curve\.

More broadly, the present construction may be viewed as a first step toward analogous representations for more general manifolds\. Extensions to higher\-dimensional curves and surfaces appear conceptually natural, but, as indicated in[Remark˜3\.10](https://arxiv.org/html/2606.06505#S3.Thmremark10),[Remark˜3\.11](https://arxiv.org/html/2606.06505#S3.Thmremark11), and[Section˜4\.3\.1](https://arxiv.org/html/2606.06505#S4.SS3.SSS1), they also raise substantial additional analytical, geometric, and computational questions\. In particular, as noted in[Section˜4\.3\.3](https://arxiv.org/html/2606.06505#S4.SS3.SSS3), much remains to be developed before a satisfactory differential geometric framework for*Gaussian Mixture Model*representations can be established\.

## Acknowledgments and Disclosure of Funding

Funded by: \- Project Name: SFB 1574, A Circular Factory for the Perpetual Product \- Funding Agency: Deutsche Forschungsgemeinschaft \(DFG\) \- Project ID: 471687386

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## Appendix ACanonical Curve Examples, Visual Convergence, and Split Views

The primary purpose of this appendix is to document a broad set of canonical curve examples and to show the behavior of the proposed geometric*Gaussian Mixture Model*construction under successive refinement of the polygonal discretization\.

In the main body \(Figures[1](https://arxiv.org/html/2606.06505#S4.F1)and[2](https://arxiv.org/html/2606.06505#S4.F2)\) we showed only the final*Gaussian Mixture Model**Probability Density Function*s for a selection of simple and more intricate curves\. Here, we revisit each of these curves individually and provide more detailed visualizations in a standardized format\.

The appendix is organized example by example\. For each plane curve, we include:

1. 1\.a brief definition of the parametrization and its basic geometric properties;
2. 2\.remarks explaining why the example is relevant for the present framework and what qualitative behavior is expected as the number of user defined probabilistic segments \(and*Gaussian Mixture Model*components\) increases;
3. 3\.two figures: 1. \(a\)a convergence series showing the induced*Gaussian Mixture Model*for several values ofNNon a common spatial scale; 2. \(b\)a split view in which the left panel displays the curve together with ellipses that are the representative of the covariance matrices of the*Gaussian Mixture Model*components, and the right panel shows the corresponding*Probability Density Function*heatmap\.

The examples have been chosen to cover a range of topological and geometrical situations, including smooth open curves, smooth closed curves, piecewise regular curves with corners, non regular curves, and curves having singularities or self intersections\. Taken together, they provide a qualitative picture of how the proposed representation captures local tangent and local normal structure, singular behavior, global shape, and topologically disconnectedness across different classes of planar curves\.

### A\.1Line Segment

###### Definition A\.1\(Line segment\)\.

Letv0,v1∈ℝ2∧‖v0−v1‖≠0v\_\{0\},v\_\{1\}\\in\\mathbb\{R\}^\{2\}\\land\\\|v\_\{0\}\-v\_\{1\}\\\|\\neq 0\. The line segment joiningv0v\_\{0\}andv1v\_\{1\}is the curve

α:I→ℝ2,t↦v0\+t​\(v1−v0\)\.\\alpha\\colon I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto v\_\{0\}\+t\(v\_\{1\}\-v\_\{0\}\)\.\(26\)Since

α′​\(t\)=v1−v0,\\alpha^\{\\prime\}\(t\)=v\_\{1\}\-v\_\{0\},\(27\)the curve has constant tangent direction and constant speed

‖α′​\(t\)‖=‖v1−v0‖\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\\|v\_\{1\}\-v\_\{0\}\\\|\.\(28\)Its arc length is therefore

ℓ=∫I‖α′​\(t\)‖​dt=‖v1−v0‖\.\\ell=\\int\_\{I\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=\\\|v\_\{1\}\-v\_\{0\}\\\|\.\(29\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x3.png)Figure 3:Convergence of the*Gaussian Mixture Model*representation for the line segment, a straight regular line segment fromx1=\(−1,−1\)x\_\{1\}=\(\-1,\-1\)tox2=\(1,1\)x\_\{2\}=\(1,1\), as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x4.png)Figure 4:Split view for the line segment, a straight regular line segment fromx1=\(−1,−1\)x\_\{1\}=\(\-1,\-1\)tox2=\(1,1\)x\_\{2\}=\(1,1\): cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.2Circle

###### Definition A\.2\(Circle\)\.

Letr∈ℝ\>0r\\in\\mathbb\{R\}\_\{\>0\}\. The circle of radiusrrcentered at zero is the curve

α:\[0,2​π\)→ℝ2,t↦\(r​cos⁡t,r​sin⁡t\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(r\\cos t,r\\sin t\)\.\(30\)Its derivative is

α′:\(0,2​π\)→ℝ2,t↦\(−r​sin⁡t,r​cos⁡t\)\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(\-r\\sin t,r\\cos t\)\(31\)and therefore

‖α′​\(t\)‖=rfor all​t∈\(0,1\)\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=r\\qquad\\text\{for all \}t\\in\(0,1\)\.\(32\)Thus the circle is smooth and regular, has constant speed, and its arc length is

ℓ=∫02​π‖α′​\(t\)‖​𝑑t=2​π​r\.\\ell=\\int\_\{0\}^\{2\\pi\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,dt=2\\pi r\.\(33\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x5.png)Figure 5:Convergence of the*Gaussian Mixture Model*representation for the circle, a smooth regular closed curve, centered at zero of radiusr=1r=1, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x6.png)Figure 6:Split view for the circle, a smooth regular closed curve, centered at zero of radiusr=1r=1: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.3Parabola

###### Definition A\.3\(Parabola\)\.

LetI=\[tmin,tmax\]⊂ℝI=\[t\_\{\\min\},t\_\{\\max\}\]\\subset\\mathbb\{R\}\. The parabola segment is the mapping

α:I→ℝ2,t↦\(t,t2\)\.\\alpha:I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(t,\\,t^\{2\}\)\.\(34\)Its derivative is the mapping

α′:I→ℝ2,t↦\(1,2​t\)\.\\alpha^\{\\prime\}:I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(1,\\,2t\)\.\(35\)Hence

‖α′​\(t\)‖=1\+4​t2\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\sqrt\{1\+4t^\{2\}\}\.\(36\)The arc length onIIis

ℓ=∫tmintmax‖α′​\(t\)‖​𝑑t=∫tmintmax1\+4​t2​𝑑t\.\\ell=\\int\_\{t\_\{\\min\}\}^\{t\_\{\\max\}\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,dt=\\int\_\{t\_\{\\min\}\}^\{t\_\{\\max\}\}\\sqrt\{1\+4t^\{2\}\}\\,dt\.\(37\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x7.png)Figure 7:Convergence of the*Gaussian Mixture Model*representation for the parabola segment onI=\[−1,1\]I=\[\-1,1\], a smooth open graph type curve, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x8.png)Figure 8:Split view for the parabola segment onI=\[−1,1\]I=\[\-1,1\], a smooth open graph type curve: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.4Graph of a Function

###### Definition A\.4\(Curve Equivalent of Graph of a Function\)\.

ForX=I,Y⊂ℝX=I,Y\\subset\\mathbb\{R\}let the mapping of interest be

f:X→Y,t↦f​\(t\)f\\colon X\\to Y,\\qquad t\\mapsto f\(t\)\(38\)whereIIis an interval\. The associated plane curve is the mapping

α:I→ℝ2,t↦\(t,f​\(t\)\)\.\\alpha\\colon I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(t,\\,f\(t\)\)\.\(39\)Its derivative is the mapping

α′:I→ℝ2,t↦\(1,f′​\(t\)\)\.\\alpha^\{\\prime\}\\colon I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(1,f^\{\\prime\}\(t\)\)\.\(40\)Hence

‖α′​\(t\)‖=1\+f′​\(t\)2\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\sqrt\{1\+f^\{\\prime\}\(t\)^\{2\}\}\.\(41\)The arc length is

ℓ=∫I‖α′​\(t\)‖​dt=∫I1\+f′​\(t\)2​dt\.\\ell=\\int\_\{I\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\mathrm\{d\}t=\\int\_\{I\}\\sqrt\{1\+f^\{\\prime\}\(t\)^\{2\}\}\\mathrm\{d\}t\.\(42\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x9.png)Figure 9:Convergence of the*Gaussian Mixture Model*representation for a bijective strict monotone function, a smooth graph type curve onI=\[0,1\]I=\[0,1\], as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x10.png)Figure 10:Split view for the nonlinear measurement curve, a smooth graph type curve onI=\[0,1\]I=\[0,1\]: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.5Mirrored S Shaped Curve

###### Definition A\.5\(Mirrored S Shaped\)\.

LetT∈ℝ\>0T\\in\\mathbb\{R\}\_\{\>0\}, then the curve

α:\[−T,T\]→ℝ2,t↦\(t,tanh⁡\(−4​t\)\)\.\\alpha\\colon\[\-T,T\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(t,\\,\\tanh\(\-4t\)\)\.\(43\)will look like a mirrored S\. Its derivative is

α′:\(−T,T\)→ℝ2,t↦\(1,−4​sech2​\(4​t\)\),\\alpha^\{\\prime\}\\colon\(\-T,T\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(1,\\,\-4\\,\\mathrm\{sech\}^\{2\}\(4t\)\\bigr\),\(44\)and therefore

‖α′​\(t\)‖=1\+16​sech4​\(4​t\)\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\sqrt\{1\+16\\,\\mathrm\{sech\}^\{4\}\(4t\)\}\.\(45\)Since the first component ofα′​\(t\)\\alpha^\{\\prime\}\(t\)is identically equal to11, the curve is regular for allt∈\(−1,1\)t\\in\(\-1,1\)\. Its arc length is

ℓ=∫−TT‖α′​\(t\)‖​dt=∫−TT1\+16​sech4​\(4​t\)​dt\.\\ell=\\int\_\{\-T\}^\{T\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=\\int\_\{\-T\}^\{T\}\\sqrt\{1\+16\\,\\mathrm\{sech\}^\{4\}\(4t\)\}\\,\\mathrm\{d\}t\.\(46\)The second derivative is

α′′:\[−T,T\]→ℝ2,t↦\(0,32​sech2​\(4​t\)​tanh⁡\(4​t\)\)\.\\alpha^\{\\prime\\prime\}\\colon\[\-T,T\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(0,\\,32\\,\\mathrm\{sech\}^\{2\}\(4t\)\\tanh\(4t\)\\bigr\)\.\(47\)Thus, the curve is smooth and regular, with a central transition region neart=0t=0where the second derivative changes sign\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x11.png)Figure 11:Convergence of the*Gaussian Mixture Model*representation for the mirrored S\-shapedtanh\\tanhbased curve, a smooth regular graph type curve on\[−1,1\]\[\-1,1\], as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x12.png)Figure 12:Split view for the mirrored S\-shapedtanh\\tanhcurve, a smooth regular graph type curve on\[−1,1\]\[\-1,1\]: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.6Downward Logarithmic Spiral

###### Definition A\.6\(Downward Logarithmic Spiral\)\.

Leta\>0a\>0andb∈ℝ<0b\\in\\mathbb\{R\}\_\{<0\}, and letI=\[tmin,tmax\]⊂ℝI=\[t\_\{\\min\},t\_\{\\max\}\]\\subset\\mathbb\{R\}\. The downward logarithmic spiral is the mapping

α:I→ℝ2,t↦\(a​eb​t​cos⁡t,a​eb​t​sin⁡t\)\.\\alpha\\colon I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(ae^\{bt\}\\cos t,\\,ae^\{bt\}\\sin t\\bigr\)\.\(48\)Its derivative is the mapping

α′:I→ℝ2,t↦\(a​eb​t​\(b​cos⁡t−sin⁡t\),a​eb​t​\(b​sin⁡t\+cos⁡t\)\)\.\\alpha^\{\\prime\}\\colon I\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(ae^\{bt\}\(b\\cos t\-\\sin t\),\\,ae^\{bt\}\(b\\sin t\+\\cos t\)\\bigr\)\.\(49\)Therefore

‖α′​\(t\)‖=a​eb​t​1\+b2\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=ae^\{bt\}\\sqrt\{1\+b^\{2\}\}\.\(50\)Hence the curve is smooth and regular onII, and its arc length is

ℓ=∫I‖α′​\(t\)‖​dt=a​1\+b2​∫Ieb​t​dt\.\\ell=\\int\_\{I\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=a\\sqrt\{1\+b^\{2\}\}\\int\_\{I\}e^\{bt\}\\,\\mathrm\{d\}t\.\(51\)In particular, sinceIIis compact andb∈ℝ<0b\\in\\mathbb\{R\}\_\{<0\}, the curve is rectifiable\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x13.png)Figure 13:Convergence of the*Gaussian Mixture Model*representation for the downward logarithmic spiral, a smooth regular curve with parametersa=0\.1a=0\.1,b=0\.2b=0\.2onI=\[0,6​π\]I=\[0,6\\pi\], as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x14.png)Figure 14:Split view for the logarithmic spiral, a smooth regular spiral curve with parametersa=0\.1a=0\.1,b=0\.2b=0\.2onI=\[0,6​π\]I=\[0,6\\pi\]: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.7Semicubical Cusp

###### Definition A\.7\(Semicubical Cusp\)\.

LetT∈ℝ\>0T\\in\\mathbb\{R\}\_\{\>0\}\. The semicubical cusp is the mapping

α:\[−T,T\]→ℝ2,t↦\(t2,t3\)\.\\alpha\\colon\[\-T,T\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(t^\{2\},t^\{3\}\)\.\(52\)Its derivative is the mapping

α′:\[−T,T\]→ℝ2,t↦\(2​t,3​t2\),\\alpha^\{\\prime\}\\colon\[\-T,T\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(2t,3t^\{2\}\),\(53\)therefore

‖α′​\(t\)‖=4​t2\+9​t4\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\sqrt\{4t^\{2\}\+9t^\{4\}\}\.\(54\)The curve is regular fort≠0t\\neq 0and has a cusp att=0t=0, where

α′​\(0\)=\(0,0\)\.\\alpha^\{\\prime\}\(0\)=\(0,0\)\.\(55\)Its arc length is

ℓ=∫−11‖α′​\(t\)‖​dt=∫−114​t2\+9​t4​dt\.\\ell=\\int\_\{\-1\}^\{1\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=\\int\_\{\-1\}^\{1\}\\sqrt\{4t^\{2\}\+9t^\{4\}\}\\,\\mathrm\{d\}t\.\(56\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x15.png)Figure 15:Convergence of the*Gaussian Mixture Model*representation for the semicubical cusp, a non regular curve on\[−1,1\]\[\-1,1\]with an isolated cusp att=0t=0, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x16.png)Figure 16:Split view for the semicubical cusp, a non regular curve on\[−1,1\]\[\-1,1\]with an isolated cusp att=0t=0: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.8Cycloid

###### Definition A\.8\(Cycloid\)\.

Letr∈ℝ\>0,tmin,tmax∈ℝr\\in\\mathbb\{R\}\_\{\>0\},t\_\{\\mathrm\{min\}\},t\_\{\\mathrm\{max\}\}\\in\\mathbb\{R\}\. The cycloid is the mapping

α:\[tmin,tmax\]→ℝ2,t↦\(r​\(t−sin⁡t\),r​\(1−cos⁡t\)\)\.\\alpha\\colon\[t\_\{\\mathrm\{min\}\},t\_\{\\mathrm\{max\}\}\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(r\(t\-\\sin t\),\\,r\(1\-\\cos t\)\\bigr\)\.\(57\)Its derivative is the mapping

α′:\(tmin,tmax\)→ℝ2,t↦\(r​\(1−cos⁡t\),r​sin⁡t\)\.\\alpha^\{\\prime\}\\colon\(t\_\{\\mathrm\{min\}\},t\_\{\\mathrm\{max\}\}\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(r\(1\-\\cos t\),\\,r\\sin t\\bigr\)\.\(58\)Therefore

‖α′​\(t\)‖=2​r​\|sin⁡\(t2\)\|\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=2r\\left\|\\sin\\\!\\left\(\\frac\{t\}\{2\}\\right\)\\right\|\.\(59\)The curve is regular for allt∈\[tmin,tmax\]∖\{2​π​k:k∈ℤ\}t\\in\[t\_\{\\mathrm\{min\}\},t\_\{\\mathrm\{max\}\}\]\\setminus\\\{2\\pi k\\colon k\\in\\mathbb\{Z\}\\\}, and it has cusp points whenevert=2​π​kt=2\\pi klies in\[tmin,tmax\]\[t\_\{\\mathrm\{min\}\},t\_\{\\mathrm\{max\}\}\], since then

α′​\(2​π​k\)=\(0,0\)\.\\alpha^\{\\prime\}\(2\\pi k\)=\(0,0\)\.\(60\)Hence its arc length is

ℓ=∫tmintmax2​r​\|sin⁡\(t2\)\|​dt\.\\ell=\\int\_\{t\_\{\\mathrm\{min\}\}\}^\{t\_\{\\mathrm\{max\}\}\}2r\\left\|\\sin\\\!\\left\(\\frac\{t\}\{2\}\\right\)\\right\|\\mathrm\{d\}t\.\(61\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x17.png)Figure 17:Convergence of the*Gaussian Mixture Model*representation for the cycloid, a curve with cusp singularities and with the parametersr=1,tmin=0,tmax=2​πr=1,t\_\{\\mathrm\{min\}\}=0,t\_\{\\mathrm\{max\}\}=2\\pi, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x18.png)Figure 18:Split view for the cycloid, a curve with cusp singularities and radiusr=1r=1over one archT=2​πT=2\\pi: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.9Square

###### Definition A\.9\(Square\)\.

LetL∈ℝ\>0L\\in\\mathbb\{R\}\_\{\>0\}, and

v1=\(−L2,−L2\),v2=\(L2,−L2\),v3=\(L2,L2\),v4=\(−L2,L2\)\.v\_\{1\}=\\left\(\-\\frac\{L\}\{2\},\-\\frac\{L\}\{2\}\\right\),\\qquad v\_\{2\}=\\left\(\\frac\{L\}\{2\},\-\\frac\{L\}\{2\}\\right\),\\qquad v\_\{3\}=\\left\(\\frac\{L\}\{2\},\\frac\{L\}\{2\}\\right\),\\qquad v\_\{4\}=\\left\(\-\\frac\{L\}\{2\},\\frac\{L\}\{2\}\\right\)\.\(62\)The boundary of the axis aligned square of side lengthLLcentered at the origin is the closed polygonal curve through the verticesv1,v2,v3,v4,v1v\_\{1\},v\_\{2\},v\_\{3\},v\_\{4\},v\_\{1\}\. A convenient piecewise parametrization is the mapping

α:\[0,4\]→ℝ2,t↦\{\(−L2\+L​t,−L2\),t∈\[0,1\],\(L2,−L2\+L​\(t−1\)\),t∈\[1,2\],\(L2−L​\(t−2\),L2\),t∈\[2,3\],\(−L2,L2−L​\(t−3\)\),t∈\[3,4\]\.\\alpha\\colon\[0,4\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\begin\{cases\}\\left\(\-\\dfrac\{L\}\{2\}\+Lt,\\,\-\\dfrac\{L\}\{2\}\\right\),&t\\in\[0,1\],\\\\\[6\.45831pt\] \\left\(\\dfrac\{L\}\{2\},\\,\-\\dfrac\{L\}\{2\}\+L\(t\-1\)\\right\),&t\\in\[1,2\],\\\\\[6\.45831pt\] \\left\(\\dfrac\{L\}\{2\}\-L\(t\-2\),\\,\\dfrac\{L\}\{2\}\\right\),&t\\in\[2,3\],\\\\\[6\.45831pt\] \\left\(\-\\dfrac\{L\}\{2\},\\,\\dfrac\{L\}\{2\}\-L\(t\-3\)\\right\),&t\\in\[3,4\]\.\\end\{cases\}\(63\)Its derivative on each open subinterval is given by

α′​\(t\)=\{\(L,0\),t∈\(0,1\),\(0,L\),t∈\(1,2\),\(−L,0\),t∈\(2,3\),\(0,−L\),t∈\(3,4\)\.\\alpha^\{\\prime\}\(t\)=\\begin\{cases\}\(L,0\),&t\\in\(0,1\),\\\\\[6\.45831pt\] \(0,L\),&t\\in\(1,2\),\\\\\[6\.45831pt\] \(\-L,0\),&t\\in\(2,3\),\\\\\[6\.45831pt\] \(0,\-L\),&t\\in\(3,4\)\.\\end\{cases\}\(64\)Hence

‖α′​\(t\)‖=L\\\|\\alpha^\{\\prime\}\(t\)\\\|=L\(65\)for allt∈\(0,1\)∪\(1,2\)∪\(2,3\)∪\(3,4\)t\\in\(0,1\)\\cup\(1,2\)\\cup\(2,3\)\\cup\(3,4\)\. Thus, the curve is regular on each open edge, but it is not differentiable at the corner parameterst=1,2,3t=1,2,3and at the identified endpointst=0,4t=0,4\. Its total arc length is

ℓ=∑k=14∫k−1k‖α′​\(t\)‖​dt=4​L\.\\ell=\\sum\_\{k=1\}^\{4\}\\int\_\{k\-1\}^\{k\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=4L\.\(66\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x19.png)Figure 19:Convergence of the*Gaussian Mixture Model*representation for the square curve, a closed polygonal curve of side lengthL=1L=1, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x20.png)Figure 20:Split view for the square boundary, a closed polygonal curve of side lengthL=1L=1: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.10Astroid

###### Definition A\.10\(Astroid\)\.

Leta∈ℝ\>0a\\in\\mathbb\{R\}\_\{\>0\}\. The astroid is the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​cos3⁡t,a​sin3⁡t\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\\cos^\{3\}t,\\,a\\sin^\{3\}t\\bigr\)\.\(67\)Its derivative is the mapping

α′:\(0,2​π\)→ℝ2,t↦\(−3​a​cos2⁡t​sin⁡t,3​a​sin2⁡t​cos⁡t\)\.\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(\-3a\\cos^\{2\}t\\sin t,\\,3a\\sin^\{2\}t\\cos t\\bigr\)\.\(68\)Therefore

‖α′​\(t\)‖=3​a​\|sin⁡t​cos⁡t\|\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=3a\\,\|\\sin t\\cos t\|\.\(69\)The curve is regular for all

t∈\[0,2​π\]∖\{0,π2,π,3​π2,2​π\},t\\in\[0,2\\pi\]\\setminus\\left\\\{0,\\frac\{\\pi\}\{2\},\\pi,\\frac\{3\\pi\}\{2\},2\\pi\\right\\\},\(70\)and it fails to be regular at the parameter values

t∈\{0,π2,π,3​π2,2​π\},t\\in\\left\\\{0,\\frac\{\\pi\}\{2\},\\pi,\\frac\{3\\pi\}\{2\},2\\pi\\right\\\},\(71\)where

α′​\(t\)=\(0,0\)\.\\alpha^\{\\prime\}\(t\)=\(0,0\)\.\(72\)These parameter values correspond to the four cusp singularities of the astroid\. Its arc length is

ℓ=∫02​π‖α′​\(t\)‖​dt=∫02​π3​a​\|sin⁡t​cos⁡t\|​dt=6​a\.\\ell=\\int\_\{0\}^\{2\\pi\}\\\|\\alpha^\{\\prime\}\(t\)\\\|\\,\\mathrm\{d\}t=\\int\_\{0\}^\{2\\pi\}3a\\,\|\\sin t\\cos t\|\\,\\mathrm\{d\}t=6a\.\(73\)Its image is a closed algebraic curve with four cusps located at

\(±a,0\),\(0,±a\)\.\(\\pm a,0\),\\qquad\(0,\\pm a\)\.\(74\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x21.png)Figure 21:Convergence of the*Gaussian Mixture Model*representation for the astroid, a closed algebraic curve with four cusps and parametera=1a=1, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x22.png)Figure 22:Split view for the astroid, a closed algebraic curve with four cusps and parametera=1a=1: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.11Cardioid

###### Definition A\.11\(Cardioid\)\.

Leta∈ℝ\>0a\\in\\mathbb\{R\}\_\{\>0\}\. A cardioid may be parametrized by the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​\(2​cos⁡t−cos⁡\(2​t\)\),a​\(2​sin⁡t−sin⁡\(2​t\)\)\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\(2\\cos t\-\\cos\(2t\)\),\\,a\(2\\sin t\-\\sin\(2t\)\)\\bigr\)\.\(75\)Its derivative is the mapping

α′:\(0,2​π\)→ℝ2,t↦\(2​a​\(sin⁡\(2​t\)−sin⁡t\),2​a​\(cos⁡t−cos⁡\(2​t\)\)\)\.\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(2a\(\\sin\(2t\)\-\\sin t\),\\,2a\(\\cos t\-\\cos\(2t\)\)\\bigr\)\.\(76\)Using the identities

sin⁡\(2​t\)=2​sin⁡t​cos⁡t,cos⁡\(2​t\)=2​cos2⁡t−1,\\sin\(2t\)=2\\sin t\\cos t,\\qquad\\cos\(2t\)=2\\cos^\{2\}t\-1,\(77\)we obtain

α′​\(0\)=\(0,0\)\.\\alpha^\{\\prime\}\(0\)=\(0,0\)\.\(78\)Moreover,

‖α′​\(t\)‖=4​a​\|sin⁡\(t2\)\|\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=4a\\left\|\\sin\\\!\\left\(\\frac\{t\}\{2\}\\right\)\\right\|\.\(79\)Hence the curve is regular for allt∈\(0,2​π\]t\\in\(0,2\\pi\], and it fails to be regular att=0t=0, which corresponds to the cusp of the cardioid\. Thus, the cardioid is a closed planar curve with a single cusp singularity\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x23.png)Figure 23:Convergence of the*Gaussian Mixture Model*representation for the cardioid, a closed planar curve with a single cusp and parametera=1a=1, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x24.png)Figure 24:Split view for the cardioid, a closed planar curve with a single cusp and parametera=1a=1: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.12Lemniscate

###### Definition A\.12\(Lemniscate\)\.

Leta∈ℝ\>0a\\in\\mathbb\{R\}\_\{\>0\}\. A convenient parametrization of a lemniscate type curve is the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​cos⁡t,a​sin⁡t​cos⁡t\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\\cos t,\\,a\\sin t\\cos t\\bigr\)\.\(80\)Its derivative is the mapping

α′:\(0,2​π\)→ℝ2,t↦\(−a​sin⁡t,a​cos⁡\(2​t\)\)\.\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(\-a\\sin t,\\,a\\cos\(2t\)\\bigr\)\.\(81\)Therefore

‖α′​\(t\)‖=a​sin2⁡t\+cos2⁡\(2​t\)\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=a\\sqrt\{\\sin^\{2\}t\+\\cos^\{2\}\(2t\)\}\.\(82\)The curve is smooth on\(0,2​π\)\(0,2\\pi\)\. Moreover, sincesin⁡t\\sin tandcos⁡\(2​t\)\\cos\(2t\)do not vanish simultaneously, it is regular on\(0,2​π\)\(0,2\\pi\)\. Furthermore,

α​\(π2\)=α​\(3​π2\)=\(0,0\),\\alpha\\\!\\left\(\\frac\{\\pi\}\{2\}\\right\)=\\alpha\\\!\\left\(\\frac\{3\\pi\}\{2\}\\right\)=\(0,0\),\(83\)so the image has a infinity shape structure with a self intersection at the origin\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x25.png)Figure 25:Convergence of the*Gaussian Mixture Model*representation for the lemniscate, a smooth regular infinity shaped closed curve with a self intersection and with the parametera=1a=1, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x26.png)Figure 26:Split view for the lemniscate, a smooth regular infinity shaped closed curve with a self intersection and with the parametera=1a=1: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.13Rose

###### Definition A\.13\(Rose\)\.

Leta∈ℝ\>0a\\in\\mathbb\{R\}\_\{\>0\}and letk∈ℤ∖\{0\}k\\in\\mathbb\{Z\}\\setminus\\\{0\\\}\. The \(parametrized\) rose curve with parameterkkis the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​cos⁡\(k​t\)​cos⁡t,a​cos⁡\(k​t\)​sin⁡t\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\\cos\(kt\)\\cos t,\\,a\\cos\(kt\)\\sin t\\bigr\)\.\(84\)Its derivative is the mapping

α′:\(0,2​π\)→ℝ2,t↦\(a​\(−k​sin⁡\(k​t\)​cos⁡t−cos⁡\(k​t\)​sin⁡t\),a​\(−k​sin⁡\(k​t\)​sin⁡t\+cos⁡\(k​t\)​cos⁡t\)\)\.\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\(\-k\\sin\(kt\)\\cos t\-\\cos\(kt\)\\sin t\),\\,a\(\-k\\sin\(kt\)\\sin t\+\\cos\(kt\)\\cos t\)\\bigr\)\.\(85\)Therefore,

‖α′​\(t\)‖=a​k2​sin2⁡\(k​t\)\+cos2⁡\(k​t\)\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=a\\sqrt\{k^\{2\}\\sin^\{2\}\(kt\)\+\\cos^\{2\}\(kt\)\}\.\(86\)Sincek2​sin2⁡\(k​t\)\+cos2⁡\(k​t\)\>0k^\{2\}\\sin^\{2\}\(kt\)\+\\cos^\{2\}\(kt\)\>0for allt∈\[0,2​π\)t\\in\[0,2\\pi\), the curve is smooth and regular on\(0,2​π\)\(0,2\\pi\)\.

The image ofα\\alphais the classical rose curve\. Its number of petals is:

- •\|k\|\|k\|petals ifkkis odd;
- •2​\|k\|2\|k\|petals ifkkis even\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x27.png)Figure 27:Convergence of the*Gaussian Mixture Model*representation for the rose curve with parametersa=1a=1andk=5k=5, a smooth regular closed curve with multiple self intersections, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x28.png)Figure 28:Split view for the rose curve with parametersa=1a=1andk=5k=5, a smooth regular closed curve with multiple self intersections: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.14Yin\-Yang Curve

###### Definition A\.14\(Yin\-Yang curve\)\.

A simple analytic Yin\-Yang type configuration may be described by an outer circle, an inner divider curve formed by two semicircular arcs, and two small “eye” circles, as illustrated in Figure[30](https://arxiv.org/html/2606.06505#A1.F30)\.

The outer circle of radius one centered at the origin can be constructed by the curve

αout:\[0,2​π\]→ℝ2,t↦\(cos⁡t,sin⁡t\)\.\\alpha\_\{\\mathrm\{out\}\}\\colon\[0,2\\pi\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\(\\cos t,\\,\\sin t\)\.\(87\)
The divider between the two halves of the Yin\-Yang symbol is realized by two semicircles of radius12\\tfrac\{1\}\{2\}, one centered at\(0,12\)\(0,\\tfrac\{1\}\{2\}\)and one at\(0,−12\)\(0,\-\\tfrac\{1\}\{2\}\), glued together at their endpoints on the outer circle\.

The upper inner semicircle \(center\(0,12\)\(0,\\tfrac\{1\}\{2\}\), radius12\\tfrac\{1\}\{2\}\) can be constructed by the curve

α\+:\(π2,3​π2\]→ℝ2,t↦\(12​cos⁡t,12\+12​sin⁡t\)\.\\alpha\_\{\+\}\\colon\\left\(\\frac\{\\pi\}\{2\},\\frac\{3\\pi\}\{2\}\\right\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\left\(\\frac\{1\}\{2\}\\cos t,\\,\\frac\{1\}\{2\}\+\\frac\{1\}\{2\}\\sin t\\right\)\.\(88\)The lower inner semicircle \(center\(0,−12\)\(0,\-\\tfrac\{1\}\{2\}\), radius12\\tfrac\{1\}\{2\}\) can be constructed by the curve

α−:\(−π2,π2\)→ℝ2,t↦\(12​cos⁡t,−12\+12​sin⁡t\)\.\\alpha\_\{\-\}\\colon\\left\(\-\\frac\{\\pi\}\{2\},\\frac\{\\pi\}\{2\}\\right\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\left\(\\frac\{1\}\{2\}\\cos t,\\,\-\\frac\{1\}\{2\}\+\\frac\{1\}\{2\}\\sin t\\right\)\.\(89\)
The two “eyes” are small circles of radiusreye\>0r\_\{\\mathrm\{eye\}\}\>0, centered at the midpoints of the inner semicircles can be constructed by the curves

αeye,\+\\displaystyle\\alpha\_\{\\mathrm\{eye\},\+\}:\[0,2​π\)→ℝ2,\\displaystyle\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},t\\displaystyle t↦\(reye​cos⁡t,12\+reye​sin⁡t\),\\displaystyle\\mapsto\\bigl\(r\_\{\\mathrm\{eye\}\}\\cos t,\\,\\tfrac\{1\}\{2\}\+r\_\{\\mathrm\{eye\}\}\\sin t\\bigr\),\(90\)αeye,−\\displaystyle\\alpha\_\{\\mathrm\{eye\},\-\}:\[0,2​π\)→ℝ2,\\displaystyle\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},t\\displaystyle t↦\(reye​cos⁡t,−12\+reye​sin⁡t\)\.\\displaystyle\\mapsto\\bigl\(r\_\{\\mathrm\{eye\}\}\\cos t,\\,\-\\tfrac\{1\}\{2\}\+r\_\{\\mathrm\{eye\}\}\\sin t\\bigr\)\.\(91\)In the configuration shown in Figure[30](https://arxiv.org/html/2606.06505#A1.F30), the eye radius is chosen such that the eyes lie strictly inside the corresponding semicircles \(i\.e\.0<reye<120<r\_\{\\mathrm\{eye\}\}<\\tfrac\{1\}\{2\}\)\.

The complete Yin\-Yang type boundary is the union of the boundaries of the outer circle, the upper and the lower semicircles, and the two eyes:

Γ=Γαout∪Γα−∪Γα\+∪Γαeye,\+∪Γαeye,−\.\\Gamma=\\Gamma\_\{\\alpha\_\{\\mathrm\{out\}\}\}\\cup\\Gamma\_\{\\alpha\_\{\-\}\}\\cup\\Gamma\_\{\\alpha\_\{\+\}\}\\cup\\Gamma\_\{\\alpha\_\{\\mathrm\{eye\},\+\}\}\\cup\\Gamma\_\{\\alpha\_\{\\mathrm\{eye\},\-\}\}\.\(92\)

![Refer to caption](https://arxiv.org/html/2606.06505v1/x29.png)Figure 29:Convergence of the*Gaussian Mixture Model*representation for the Yin\-Yang curve configuration, a compound boundary composed of circular arcs with outer radius one and inner semicircles of radius1/21/2, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x30.png)Figure 30:Split view for the Yin\-Yang curve configuration, a compound boundary composed of circular arcs with outer radius11and inner semicircles of radius1/21/2: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.15Lissajous Curve

###### Definition A\.15\(Lissajous curve\)\.

Leta,b∈ℝ\>0,p,q∈ℕ∖\{0\},δ∈ℝa,b\\in\\mathbb\{R\}\_\{\>0\},p,q\\in\\mathbb\{N\}\\setminus\\\{0\\\},\\delta\\in\\mathbb\{R\}\. A planar Lissajous curve is the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​sin⁡\(p​t\+δ\),b​sin⁡\(q​t\)\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\\sin\(pt\+\\delta\),\\,b\\sin\(qt\)\\bigr\)\.\(93\)Its derivative is the mapping

α′:\(0,2​π\)→ℝ2,t↦\(a​p​cos⁡\(p​t\+δ\),b​q​cos⁡\(q​t\)\)\.\\alpha^\{\\prime\}\\colon\(0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(ap\\cos\(pt\+\\delta\),\\,bq\\cos\(qt\)\\bigr\)\.\(94\)Therefore

‖α′​\(t\)‖=a2​p2​cos2⁡\(p​t\+δ\)\+b2​q2​cos2⁡\(q​t\)\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=\\sqrt\{a^\{2\}p^\{2\}\\cos^\{2\}\(pt\+\\delta\)\+b^\{2\}q^\{2\}\\cos^\{2\}\(qt\)\}\.\(95\)Hence the curve is smooth on\(0,2​π\)\(0,2\\pi\)and regular at all parameter values for which not bothcos⁡\(p​t\+δ\)\\cos\(pt\+\\delta\)andcos⁡\(q​t\)\\cos\(qt\)vanish simultaneously\.

In the implementation, we use the special case

a=p,b=q,a=p,\\quad b=q,\(96\)so that both amplitude and frequency in each coordinate are controlled by the same integer parameters\. The example used in the figures corresponds top=3p=3,q=2q=2, andδ=π/4\\delta=\\pi/4\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x31.png)Figure 31:Convergence of the*Gaussian Mixture Model*representation for a Lissajous curve with parameters\(a,b,p,q,δ\)=\(5,3,5,3,π/4\)\(a,b,p,q,\\delta\)=\(5,3,5,3,\\pi/4\), a smooth closed curve with oscillatory structure and multiple self intersections, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x32.png)Figure 32:Split view for the Lissajous curve with parameters\(a,b,p,q,δ\)=\(5,3,5,3,π/4\)\(a,b,p,q,\\delta\)=\(5,3,5,3,\\pi/4\), a smooth closed curve with oscillatory structure and multiple self intersections: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.16Archimedean Spiral

###### Definition A\.16\(Archimedean spiral\)\.

Leta,T∈ℝ\>0a,T\\in\\mathbb\{R\}\_\{\>0\}\. The Archimedean spiral is the mapping

α:\[0,T\]→ℝ2,t↦\(a​t​cos⁡t,a​t​sin⁡t\)\.\\alpha\\colon\[0,T\]\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(at\\cos t,\\,at\\sin t\\bigr\)\.\(97\)Its derivative is the mapping

α′:\(0,T→ℝ2,t↦\(a\(cost−tsint\),a\(sint\+tcost\)\)\.\\alpha^\{\\prime\}\\colon\(0,T\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\(\\cos t\-t\\sin t\),\\,a\(\\sin t\+t\\cos t\)\\bigr\)\.\(98\)Therefore

‖α′​\(t\)‖=a​1\+t2\.\\\|\\alpha^\{\\prime\}\(t\)\\\|=a\\sqrt\{1\+t^\{2\}\}\.\(99\)Hence the curve is smooth and regular on\[0,T\]\[0,T\]\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x33.png)Figure 33:Convergence of the*Gaussian Mixture Model*representation for the Archimedean spiral, a smooth regular spiral curve with parametera=0\.1a=0\.1onI=\[0,12​π\]I=\[0,12\\pi\], as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x34.png)Figure 34:Split view for the Archimedean spiral, a smooth regular spiral curve with parametera=0\.1a=0\.1on\[0,12​π\]\[0,12\\pi\]: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.
### A\.17Superellipse

###### Definition A\.17\(Superellipse\)\.

Leta,b,m∈ℝ\>0a,b,m\\in\\mathbb\{R\}\_\{\>0\}\. A centered superellipse \(Lamé curve\) is the mapping

α:\[0,2​π\)→ℝ2,t↦\(a​sgn⁡\(cos⁡t\)​\|cos⁡t\|2/m,b​sgn⁡\(sin⁡t\)​\|sin⁡t\|2/m\)\.\\alpha\\colon\[0,2\\pi\)\\to\\mathbb\{R\}^\{2\},\\qquad t\\mapsto\\bigl\(a\\,\\operatorname\{sgn\}\(\\cos t\)\\,\|\\cos t\|^\{2/m\},\\,b\\,\\operatorname\{sgn\}\(\\sin t\)\\,\|\\sin t\|^\{2/m\}\\bigr\)\.\(100\)Its derivative, wheneversin⁡t≠0\\sin t\\neq 0andcos⁡t≠0\\cos t\\neq 0, is given by

α′​\(t\)=\(−2​am​sgn⁡\(cos⁡t\)​\|cos⁡t\|2/m−1​sin⁡t,2​bm​sgn⁡\(sin⁡t\)​\|sin⁡t\|2/m−1​cos⁡t\)\.\\alpha^\{\\prime\}\(t\)=\\left\(\-\\frac\{2a\}\{m\}\\,\\operatorname\{sgn\}\(\\cos t\)\\,\|\\cos t\|^\{2/m\-1\}\\sin t,\\,\\frac\{2b\}\{m\}\\,\\operatorname\{sgn\}\(\\sin t\)\\,\|\\sin t\|^\{2/m\-1\}\\cos t\\right\)\.\(101\)Hence the curve is smooth on every open subinterval of\[0,2​π\)\[0,2\\pi\)on whichsin⁡t\\sin tandcos⁡t\\cos tdo not vanish\. On any such subinterval, the curve is also regular providedα′​\(t\)≠\(0,0\)\\alpha^\{\\prime\}\(t\)\\neq\(0,0\)\. Its image is a closed curve\. Form=2m=2, it reduces to an ellipse, while for other values ofmmit yields the family of Lamé curves\. Form≥1m\\geq 1, the image is convex\.

![Refer to caption](https://arxiv.org/html/2606.06505v1/x35.png)Figure 35:Convergence of the*Gaussian Mixture Model*representation for the superellipse, a Lamé curve with parametersa=2a=2,b=1b=1, andm=2m=2, as the number of componentsNNincreases\.![Refer to caption](https://arxiv.org/html/2606.06505v1/x36.png)Figure 36:Split view for the superellipse, a Lamé curve with parametersa=2a=2,b=1b=1, andm=2m=2: cover of the curve by ellipses that are the representatives of the covariances of the Gaussian components of the*Gaussian Mixture Model*representation \(left\) and the corresponding*Gaussian Mixture Model**Probability Density Function*heatmap \(right\)\.