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Entropy Minimizing Curves with Application to Flight Path Design and Clustering^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Entropy-Minimizing Curves

#### 2.1. Motivation

#### 2.2. Spatial Density of a System of Curves

#### 2.3. Further Properties of the Density

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 2.4. Minimizing the Entropy

**Proposition**

**3.**

**Proof.**

- (a)
- $\forall \eta \in [0,1],\varphi (0,\eta )={\gamma}_{i}(\eta )$.
- (b)
- $\forall (t,\eta )\in ]-a,a[\times ]0,1[,\parallel {\partial}_{\eta}\varphi (t,\eta )\parallel ={l}_{\varphi}(t)$ with ${l}_{\varphi}(t)$ the length of the curve $\eta \mapsto \varphi (t,\eta )$.
- (c)
- $\forall t\in ]-a,a[,\varphi (t,0)={\gamma}_{i}(0),\phantom{\rule{0.166667em}{0ex}}\varphi (t,1)={\gamma}_{i}(1)$.

**Proposition**

**4.**

## 3. Numerical Implementation

## 4. Conclusions and Future Work

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Puechmorel, S.; Nicol, F. Entropy Minimizing Curves with Application to Flight Path Design and Clustering. *Entropy* **2016**, *18*, 337.
https://doi.org/10.3390/e18090337

**AMA Style**

Puechmorel S, Nicol F. Entropy Minimizing Curves with Application to Flight Path Design and Clustering. *Entropy*. 2016; 18(9):337.
https://doi.org/10.3390/e18090337

**Chicago/Turabian Style**

Puechmorel, Stéphane, and Florence Nicol. 2016. "Entropy Minimizing Curves with Application to Flight Path Design and Clustering" *Entropy* 18, no. 9: 337.
https://doi.org/10.3390/e18090337