UTM与经纬度之间的转换Word格式.docx
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UTM与经纬度之间的转换Word格式.docx
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Unlikelatitudeandlongitude,thereisnophysicalframeofreferenceforUTMgrids.Latitudeisdeterminedbytheearth'
spolaraxis.Longitudeisdeterminedbytheearth'
srotation.IfyoucanseethestarsandhaveasextantandagoodclocksettoGreenwichtime,youcanfindyourlatitudeandlongitude.ButthereisnowaytodetermineyourUTMcoordinatesexceptbycalculation.
UTMgrids,ontheotherhand,arecreatedbylayingasquaregridontheearth.Thismeansthatdifferentmapswillhavedifferentgridsdependingonthedatumused(modeloftheshapeoftheearth).IsawUSmilitarymapsofGermanyshifttheirUTMgridsbyabout300meterswhenamoremoderndatumwasusedforthemaps.Also,oldWorldWarIIeramapsofEuropeapparentlyusedasinglegridforallofEuropeandgridsinsomeareasarewildlytiltedwithrespecttolatitudeandlongitude.
ThetwobasicreferencesforconvertingUTMandgeographiccoordinatesareU.S.GeologicalSurveyProfessionalPaper1395andU.S.ArmyTechnicalManualTM5-241-8(completecitationsbelow).Eachhasadvantagesanddisadvantages.
ForconvertinglatitudeandlongitudetoUTM,theArmypublicationisbetter.Itsnotationismoreconsistentandtheformulasmoreclearlylaidout,makingcodeeasiertodebug.IndefenseoftheUSGS,theirnotationisconstrainedbyspace,andtheneedtobeconsistentwithcartographicliteratureanddescriptionsofseveraldozenothermapprojectionsinthebook.
ForconvertingUTMtolatitudeandlongitude,however,theArmypublicationhasformulasthatinvolvelatitude(thequantitytobefound)andwhichrequirereverseinterpolationfromtables.HeretheUSGSpublicationistheonlygameintown.
Someextremelytinytermsthatwillnotseriouslyaffectmeter-scaleaccuracyhavebeenomitted.
ConvertingBetweenDecimalDegrees,Degrees,MinutesandSeconds,andRadians
(dd+mm/60+ss/3600)toDecimaldegrees(dd.ff)
dd=wholedegrees,mm=minutes,ss=seconds
dd.ff=dd+mm/60+ss/3600
Example:
30degrees15minutes22seconds=30+15/60+22/3600=30.2561
Decimaldegrees(dd.ff)to(dd+mm/60+ss/3600)
Forthereverseconversion,wewanttoconvertdd.fftoddmmss.Hereff=thefractionalpartofadecimaldegree.
mm=60*ff
ss=60*(fractionalpartofmm)
Useonlythewholenumberpartofmminthefinalresult.
30.2561degrees=30degrees
.2561*60=15.366minutes
.366minutes=22seconds,sothefinalresultis30degrees15minutes22seconds
Decimaldegrees(dd.ff)toRadians
Radians=(dd.ff)*pi/180
RadianstoDecimaldegrees(dd.ff)
(dd.ff)=Radians*180/pi
Degrees,MinutesandSecondstoDistance
Adegreeoflongitudeattheequatoris111.2kilometers.Aminuteis1853meters.Asecondis30.9meters.Forotherlatitudesmultiplybycos(lat).Distancesfordegrees,minutesandsecondsinlatitudeareverysimilaranddifferveryslightlywithlatitude.(Beforesatellites,observingthosedifferenceswasaprincipalmethodfordeterminingtheexactshapeoftheearth.)
ConvertingLatitudeandLongitudetoUTM
Okay,takeadeepbreath.Thiswillgetverycomplicated,butthemath,althoughtedious,isonlyalgebraandtrigonometry.Itwouldsurebeniceifsomeonewroteaspreadsheettodothis.P=pointunderconsideration
F=footofperpendicularfromPtothecentralmeridian.ThelatitudeofFiscalledthefootprintlatitude.
O=origin(onequator)
OZ=centralmeridian
LP=paralleloflatitudeofP
ZP=meridianofP
OL=k0S=meridionalarcfromequator
LF=ordinateofcurvature
OF=N=gridnorthing
FP=E=griddistancefromcentralmeridian
GN=gridnorth
C=convergenceofmeridians=anglebetweentrueandgridnorth
Anotherthingyouneedtoknowisthedatumbeingused:
Datum
EquatorialRadius,meters(a)
PolarRadius,meters(b)
Flattening(a-b)/a
Use
NAD83/WGS84
6,378,137
6,356,752.3142
1/298.257223563
Global
GRS80
6,356,752.3141
1/298.257222101
US
WGS72
6,378,135
6,356,750.5
1/298.26
NASA,DOD
Australian1965
6,378,160
6,356,774.7
1/298.25
Australia
Krasovsky1940
6,378,245
6,356,863.0
1/298.3
SovietUnion
International(1924)-Hayford(1909)
6,378,388
6,356,911.9
1/297
Globalexceptaslisted
Clake1880
6,378,249.1
6,356,514.9
1/293.46
France,Africa
Clarke1866
6,378,206.4
6,356,583.8
1/294.98
NorthAmerica
Airy1830
6,377,563.4
6,356,256.9
1/299.32
GreatBritain
Bessel1841
6,377,397.2
6,356,079.0
1/299.15
CentralEurope,Chile,Indonesia
Everest1830
6,377,276.3
6,356,075.4
1/300.80
SouthAsia
Don'
tinterpretthecharttomeanthereisradicaldisagreementabouttheshapeoftheearth.Theearthisn'
tperfectlyround,itisn'
tevenaperfectellipsoid,andslightlydifferentshapesworkbetterforsomeregionsthanfortheearthasawhole.Thetopthreearebasedonworldwidedataandaretrulyglobal.Also,youareveryunlikelytofindUTMgridsbasedonanyoftheearlierprojections.
Themostmoderndatums(jarsmyLatinistsensibilitiessincethepluralofdatuminLatinisdata,butthathasadifferentmeaningtous)areNAD83andWGS84.ThesearebasedinturnonGRS80.Differencesbetweenthethreesystemsderivemostlyfromredeterminationofstationlocationsratherthandifferencesinthedatum.Unlessyouarelocatingafirst-orderstationtosub-millimeteraccuracy(inwhichcaseyouarewaybeyondthescopeofthispage)youcanprobablyregardthemasessentiallyidentical.
IhavenoinformationontheNAD83andWGS84datums,norcanmyspreadsheetcalculatedifferencesbetweenthosedatumsandWGS84.
FormulasForConvertingLatitudeandLongitudetoUTM
TheseformulasareslightlymodifiedfromArmy(1973).Theyareaccuratetowithinlessthanameterwithinagivengridzone.Theoriginalformulasincludeanowobsoletetermthatcanbehandledmoresimply-itmerelyconvertsradianstosecondsofarc.Thattermisomittedherebutdiscussedbelow.
Symbols
lat=latitudeofpoint
long=longitudeofpoint
long0=centralmeridianofzone
k0=scalealonglong0=0.9996.Eventhoughit'
saconstant,weretainitasaseparatesymboltokeepthenumericalcoefficientssimpler,alsotoallowforsystemsthatmightuseadifferentMercatorprojection.
e=SQRT(1-b2/a2)=.08approximately.Thisistheeccentricityoftheearth'
sellipticalcross-section.
e'
2=(ea/b)2=e2/(1-e2)=.007approximately.Thequantitye'
onlyoccursinevenpowerssoitneedonlybecalculatedase'
2.
n=(a-b)/(a+b)
rho=a(1-e2)/(1-e2sin2(lat))3/2.Thisistheradiusofcurvatureoftheearthinthemeridianplane.
nu=a/(1-e2sin2(lat))1/2.Thisistheradiusofcurvatureoftheearthperpendiculartothemeridianplane.Itisalsothedistancefromthepointinquestiontothepolaraxis,measuredperpendiculartotheearth'
ssurface.
p=(long-long0)inradians(ThisdiffersfromthetreatmentintheArmyreference)
CalculatetheMeridionalArc
Sisthemeridionalarcthroughthepointinquestion(thedistancealongtheearth'
ssurfacefromtheequator).Allanglesareinradians.
S=A'
lat-B'
sin(2lat)+C'
sin(4lat)-D'
sin(6lat)+E'
sin(8lat),wherelatisinradiansand
A'
=a[1-n+(5/4)(n2-n3)+(81/64)(n4-n5)...]
B'
=(3tan/2)[1-n+(7/8)(n2-n3)+(55/64)(n4-n5)...]
C'
=(15tan2/16)[1-n+(3/4)(n2-n3)...]
D'
=(35tan3/48)[1-n+(11/16)(n2-n3)...]
E'
=(315tan4/512)[1-n...]
TheUSGSgivesthisform,whichmaybemoreappealingtosome.(TheyuseMwheretheArmyusesS)
M=a[(1-e2/4-3e4/64-5e6/256....)lat
-(3e2/8+3e4/32+45e6/1024...)sin(2lat)
+(15e4/256+45e6/1024+....)sin(4lat)
-(35e6/3072+....)sin(6lat)+....)]wherelatisinradians
Thisisthehardpart.Calculatingthearclengthofanellipseinvolvesfunctionscalledellipticintegrals,whichdon'
treducetoneatclosedformulas.Sotheyhavetoberepresentedasseries.
Allanglesareinradians.
y=northing=K1+K2p2+K3p4,where
K1=Sk0,
K2=k0nusin(lat)cos(lat)/2=k0nusin(2lat)/4
K3=[k0nusin(lat)cos3(lat)/24][(5-tan2(lat)+9e'
2cos2(lat)+4e'
4cos4(lat)]
x=easting=K4p+K5p3,where
K4=k0nucos(lat)
K5=(k0nucos3(lat)/6)[1-tan2(lat)+e'
2cos2(lat)]
Eastingxisrelativetothecentralmeridian.ForconventionalUTMeastingadd500,000meterstox.
WhattheFormulasMean
Thehardpart,allowingfortheoblatenessoftheEarth,istakencareofincalculatingS(orM).SoK1issimplythearclengthalongthecentralmeridianofthezonecorrectedbythescalefactor.Remember,thescaleisahairlessthan1inthemiddleofthezone,andahairmoreontheoutside.
AllthehigherKtermsinvolvenu,thelocalradiusofcurvature(roughlyequaltotheradiusoftheearthorroughly6,400,000m),trigfunctions,andpowersofe'
2(=.007).Sobasicallytheyarenevermuchlargerthannu.ActuallythemaximumvalueofK2isaboutnu/4(1,600,000),K3isaboutnu/24(267,000)andK5isaboutnu/6(1,070,000).Expandingtheexpressionswillshowthatthetangenttermsdon'
taffectanything.
IfwewerejusttostopwiththeK2terminthenorthing,we'
dhaveaquadraticinp.Inotherwords,we'
dapproximatetheparalleloflatitudeasaparabola.Ther
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