#
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

- De Bondt, A.; Leleu, C. 7-Year IFR Flight Movements and Service Units Forecast Update: 2014–2020; EUROCONTROL: Brussels, Belgium, 2014. [Google Scholar]
- Roussos, G.P.; Dimarogonas, D.V.; Kyriakopoulos, K.J. Distributed 3D navigation and collision avoidance for nonholonomic aircraft-like vehicles. In Proceedings of the 2009 European Control Conference, Budapest, Hungary, 23–26 August 2009.
- Hurter, C.; Ersoy, O.; Telea, A. Smooth bundling of large streaming and sequence graphs. In Proceedings of the 6th PacificVis, Sydney, Australia, 26 February–1 March 2013; pp. 41–48.
- Harman, W.H. Air Traffic Density and Distribution Measurements; No. ATC-80; Lincoln Laboratory: Lexington, MA, USA, 3 May 1979. [Google Scholar]
- Scott, D.W. Multivariate Density Estimation: Theory, Practice, and Visualization; Wiley: New York, NY, USA, 1992. [Google Scholar]
- Silverman, B.W. Density Estimation for Statistics and Data Analysis; CRC: Boca Raton, FL, USA, 1986. [Google Scholar]
- Parzen, E. On estimation of a probability density function and mode. Ann. Math. Stat.
**1962**, 33, 1065–1076. [Google Scholar] [CrossRef] - Rosenblatt, M. Remarks on some nonparametric estimates of a density function. Ann. Math. Statist.
**1956**, 27, 832–837. [Google Scholar] [CrossRef] - Epanechnikov, V.A. Non-parametric estimation of a multivariate probability density. Theory Probab. Appl.
**1969**, 14, 153–158. [Google Scholar] [CrossRef] - Michor, P.W.; Mumford, D. Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc.
**2006**, 8, 1–48. [Google Scholar] [CrossRef] - Ambrosio, L.; Gigli, N.; Savaré, G. Gradient Flows: In Metric Spaces and in the Space of Probability Measures; Springer: Basel, Switzerland, 2005. [Google Scholar]
- Sun, W.; Yuan, Y.X. Optimization Theory and Methods: Nonlinear Programming; Springer: New York, NY, USA, 2006. [Google Scholar]
- Nicol, F.; Puechmorel, S. Unsupervised aircraft trajectories clustering: A minimum entropy approach. In Proceedings of the Second International Conference on Big Data, Small Data, Linked Data and Open Data, Lisbon, Portugal, 21–25 February 2016.

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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