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entropy Article EntropyMinimizingCurveswithApplicationto FlightPathDesignandClustering† StéphanePuechmorel*andFlorenceNicol LaboratoiredeMathématiquesAppliquées, InformatiqueetAutomatiquepour l’Aérien (MAIAA),Département Scienceset Ingénieriede laNavigationAérienne (SINA),ÉcoleNationalede l’AviationCivile (ENAC),7avenue EdouardBelinCS54005,31055Toulouse,France;florence.nicol@enac.fr * Correspondence: stephane.puechmorel@enac.fr;Tel.: +33-5-6217-9503 † Thispaper isanextendedversionofourpaperpublishedin the2ndConferenceonGeometricScienceof Information,Paris,France,28–30October2015. AcademicEditors: FrédéricBarbarescoandFrankNielsen Received: 26 July2016;Accepted: 8September2016;Published: 15September2016 Abstract:Airtrafficmanagement(ATM)aimsatprovidingcompanieswithasafeandideallyoptimal aircraft trajectoryplanning. Air trafficcontrollersactonflightpaths insuchawaythatnopairof aircraft comecloser thantheregulatoryseparationnorms.With the increaseof traffic, it isexpected that thesystemwill reach its limits in thenear future: aparadigmchange inATMisplannedwith the introductionof trajectory-basedoperations. In this context, setsofwell-separatedflightpaths are computed in advance, tremendously reducing the number of unsafe situations thatmust be dealtwithbycontrollers.Unfortunately,automatedtoolsusedtogeneratesuchplanninggenerally issue trajectoriesnot complyingwithoperationalpracticesorevenflightdynamics. In thispaper, ameansofproducingrealisticair routes fromtheoutputofanautomatedtrajectorydesigntool is investigated. For that purpose, the entropyof a systemof curves is first defined, andameanof iterativelyminimizing it ispresented. Theresultingcurves formaroutenetworkthat is suitable for use inasemi-automatedATMsystemwithhumaninthe loop. Thetool introducedinthiswork is quiteversatileandmaybeappliedalso tounsupervisedclassificationofcurves: anexample isgiven forFrenchtraffic. Keywords: curve system entropy; curves manifold; curve clustering; probability distribution estimation;air trafficmanagement 1. Introduction Basedonrecent studies [1], traffic inEurope is expected togrowbyanaverageyearly rateof 2.6%,yieldinganet increaseof twomillionflightsperyearat the2020horizon. The long-termforecast givesa two-fold increase in2050over thecurrent traffic,pointingout theneedforaparadigmchange in thewayflightsaremanaged.Twomajor frameworkprograms,SESAR(SingleEuropeanSkyAir trafficmanagementResearch) inEuropeandNextgen in theU.S.havebeen launchedinorder tofirst investigatepotential solutionsandtodeploytheminasecondphase.Oneof themainchanges that the air trafficmanagement (ATM)systemwillundergo isaswitch fromairspace-basedto trajectory-based operationswithadelegationof theseparationtaskto thecrews.Within this framework, trajectories becomethebasicobjectofATM,changingthewayair trafficcontrollerswillbeworking. Inorder to alleviate theworkloadofcontrollers, trajectorieswillbeplannedseveralweeks inadvance insuch awaythatcloseencountersareminimizedandideallyremoved. For thatpurpose, several toolsare currentlybeingdeveloped;mostof themcomingfromthefieldofrobotics [2].Unfortunately,flight paths issuedbythesealgorithmsarenot tractable forahumancontrollerandneedtobesimplified. Thepurposeof thepresentwork is to introduceanautomatedprocedure that takesas inputasetof Entropy2016,9, 337 387 www.mdpi.com/journal/entropy