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排版问题

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2015MathematicalContestinModeling（MCM/ICM）SummarySheet

ThemodelofSunshadowpositioning

Inordertosolvetheproblemofsunshadowpositioning,thispapercombinesrelevantknowledgeofgeographyandastronomytoestablishsomesunshadowpositioningmodelsbasedondifferentcategoriesofdata.Thus,itachievesfastandaccuratepositioningforvideoshootinglocationanddate.

Forquestion1,firstofall,fromthepointofviewofgeography,wecalculatehourangle,latitudeangleandsolarelevationanglebasedonthreeparameterswhichhaveanimpactonshadowlengthincludingthegeographicalcoordinates,thelengthofthestraightrodandtime.Furthermore,therelationformulasbetweenshadowlengthandthethreeparametersaregiven.Asresultsindicate,theshadowlengthdecreasesandincreaseslatterwithchangeofdateandtime,andshowsdirectproportionalgrowthtothelengthofthestraightrod.Withthechangeoflongitude,shadowlengthsoarstothepeakandthenplungesto0,andsurgestothepeakandthenplummetsintheend.Itappearsslowrisingtrendtochangeoflatitude.Next,accordingtothedataofappendix1,weillustratethesunshadowdistributioncurveofstraightrodonTian'anmenSquare.

Forquestion2,onthebasisoftheanalysisforfactorseffectingshadowlengthinquestion1,webuilddoublegoalprogrammingmodelintermsofGeographyknowledge,andestablishobjectivefunctions:

.Then,undertheconstraintconditions,weapplygridapproximationalgorithmtogainthelocationcoordinatesofthestraightrod.Andthestraightrod’sgeographicalpositionthatisthemostconsistentwiththedataprovidedbythesubjectis（19.1°E,108.71°N）.ItisinDongfangHainanprovince.Thelengthoftherodis2.03m,directionangleresidualratioofthesunandshadowis1.8%and0.9%respectively.Thedeviationoftwoislittle.

Forquestion3,webuildgoalprogrammingmodelwhichissimilartothatofquestion2.Duetothedateisunknownparameter,solvingthemodelisextremelycomplicated.Inordertoacceleratecalculationspeed,weintroduceparticleswarmoptimizationalgorithm.Onthebasisofanalysisforthedataofappendix2and3,weobtainthelocationcoordinatesare（80.51°E,32.13°N）,（110.20°E,24.83°N）and（81.43°E,32.24°N）,（111.56°E,23.68°N）.Thesiteofappendix2isNgariinTibet,anditsdateis8/14or4/29.Thesiteofappendix3isWuzhouinGuangxiprovince,anditsdateis12/27or12/14.Wecanfindthattheresultoftwoalgorithmsissimilar.Butparticleswarmoptimizationalgorithmcostsshortertimethangridapproximationalgorithm.

Forquestion4,thevideodataiscollectedandpretreated.Duetotheexistenceofvideoshootingangle,wedirectlyobtaintheshadowlengthinsteadofitsactuallength.Here,weutilizeimagecorrectionmethodbasedonHoughtransformationandperspectivetransformationtorectifythestrabismusimagetogettheactualshadowlength.Then,weappliedthegottendatainthe

Content

1.Restatementoftheproblem4

1.1.Problembackground4

1.2.Problemproposing4

2.Problemanalysis5

2.1.Theanalysisofquestion15

2.2.Theanalysisofquestion25

2.3.Theanalysisofquestion35

2.4.Theanalysisofquestion45

3.Basicassumptions6

4.Symboldescription6

5.TheEstablishmentandSolutionoftheModel6

5.1.Question16

5.1.1.Modelbuilding6

5.1.2.Modelsolving8

5.1.3.Resultsanalysis10

5.2.Question2-doublegoalprogrammingmodelapproximationalgorithmbasedongrid10

5.2.1.Ddatapreprocessing10

5.2.2.Modelbuilding11

5.2.3.Solvingmodel13

5.2.4.Analysisoftheresult14

5.3.Question314

5.3.1.Setupthemodel14

5.3.2.Solvingthemodel14

5.3.4.Particleswarmoptimizationalgorithm16

5.3.5.Comprehensiveanalysisofthemodel17

5.4.Question417

5.4.1.Dataacquisitionandpreprocessing:

imagerectificationbasedonHoughtransformandperspectivetransformation17

5.4.2.Theestablishmentofmodel19

5.4.3.Modelsolving20

5.4.4.Analysisoftheresult20

6.Modelchecking20

6.1.Erroranalysis20

6.2.Visualanalysis20

7.Modelpromotion21

8.Theevaluationofthemodel22

9.Reference22

10.Appendix23

10.1.Datatable23

1.Restatementoftheproblem

1.1.Problembackground

"Daylongshadowshift"isthenaturalphenomenaeverybodyknowsinlife.Thatphraseindicatesthatthechangeofshadowonthegroundhascloserelationshipwithsolaractivity.Ourintelligentancestorsutilizedthephenomenontomakesundial,whichisoneoftheearliestandmostprecisetimingtools.Inthecontemporaryfilledwithimageinformation,howtoobtainrelevantinformationonimageshootingviatheimagedataisanimportantissueofimageanalysisdiscipline.Andtheuseofsolarlightconversiontoobtaintimeandgeographicinformationisveryconvenientandreliable.

1.2.Problemproposing

ThetechnologyofSunshadowpositioningisawayofdeterminingthevideoshootingplaceanddateviaanalyzingthesunshadowchangeoftheobjectinthevideo.Onthebasisoftheabovebackgroundandtheapplicationdemand,weputforwardthefollowingquestions:

（1）Pleaseestablishthemathematicalmodelofthechangeoftheshadowlengthandanalyzethevariationlawofeachparametereffectingtheshadowlength.Thenyoushoulddrawthechangeofshadowlengthof3-meterstraightrodonTian'anmenSquare（latitude39degrees54minutes26seconds,longitude116degrees23minutes29seconds）atOctober22,2015Beijingtime9:

00-15:

00basedonthemodelyouhavebuilt.

（2）Pleaseestablishthemathematicalmodelintermsofthedataofthesolarshadowvertexcoordinatesofafixedstraightbaronthehorizontalground.Pleaseapplyyourmodeltothevertexcoordinatedatainappendix1andgivesomepossiblelocations

（3）Pleaseestablishmathematicalmodelaccordingtothedataofthesolarshadowvertexcoordinatesofafixedstraightbaronthehorizontalgroundtodeterminethelocationanddateofthestraightbar.Thenpleaseapplyyourmodeltoshadowvertexcoordinatedatainappendix2and3andsomepossibleplaceanddate.

（4）Ifthevideohasastraightbarunderthesunshadowchange,andwehaveestimatedthatthestraightrodis2mbysomeway.Pleaseestablishamathematicalmodelofvideosites,andapplyyourmodeltogivesomepossiblelocations.Iftheshootingdateisunknown,canyoudeterminethelocationanddateoftheshootingintermsofthevideo?

2.Problemanalysis

2.1.Theanalysisofquestion1

Thequestionrequiresworkingoutthechangingcurveofstraightrodprojectionlengthinfixedlocations,givendateandrodlength.Fortheverticalstraightbaronhorizontalground,theratioofthestraightbarlengthandtheshadowlengthistangentvalueofsolarelevationangle.Therefore,itisnecessarytoknowthechangeofsolarelevationangleforthisperiod.Accesstoinformation[6]available,solaraltitudeangleandlocaltime,latitude,longitudeiscloselyrelatedto,sothatthetwoquantitiescandeterminethechangeprocessofthestraightrod.

2.2.Theanalysisofquestion2

Requirementsofthesubjectpositionofthestraightrodisobtainedaccordingtothechangeofshadowandgivethedate.Infact,itistheinverseprocessofsolvingproblemswitha.Hererodlengthandlocationareunknown,inversesolutionisverydifficult,sowillaskTransformationoftitletwointoatwoobjectiveprogrammingproblem.Whentheazimuthofthesunandshadowlengthoftheactualvalueandtheoreticalvalueoftheabsolutevalueandachievethehoursandthelongitudeandlatitudeofincomeisthepolelocation.

2.3.Theanalysisofquestion3

Thedifferencebetweenthetwoandtheproblemisthatthedateisunknown.Thefirstconsiderationisgiventothemodel.Duedateisunknown,soweneedtoconsiderall365daysayear,whichwillgreatlyincreasethecomputationtime.Fromthepointofreducingtheamountofcomputation,itisnecessarytoimprovethealgorithm.Oneofthemodernoptimizationalgorithmparticleswarmalgorithmwastakenintoaccount,andwillbeallregardedasaparticletothepositionsolution,duetothefitnessandtheobjectivefunctionassociated,selectingreasonablefitnessfunction,expecttoimprovethecomputationalefficiency.

2.4.Theanalysisofquestion4

Thedataprovidedbythevideoisgiven,thenthefirstvideodatapreprocessing.Becauseofthelongervideo,soIdonotconsiderthevideoframebyframeanalysis,buteveryonceinawhileonthevideowereanalysistoobtain,theimagegiventhelengthoftheshadowandtheshadowangleweremeasured.Tovideoshootingangle,obtaineddirectlyfromthevideoshadowlongandnottheactuallength,buttheprojectionlength.Soitisnecessarytosquintimagecorrection,actualshadowlengthisderived.Thenthemodelisusedtosolvetheproblemtwoandthreerespectively.

3.Basicassumptions

●Theannualsolaractivityisthesameasthatofthe"stars"".

●Theearthisaperfectsphere,withoutconsideringtheinfluenceofaltitude,theearthoblateness.

●Shadowfadephenomenoncausedbynolightdiffraction.

●Whenconsideringtheproblemoftheprojectionofastraightrodinasmallscale,thegroundsurfaceisabsolutelyhorizontal.

●Withoutconsideringtheinfluenceoftherevolutionoftheearth.

●Thedatagivenbythetitleistrueandreliable.

4.Symboldescription

Symbol

Description

Solarelevation

Horizontalstraightbarlength

Horizontalgroundst

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