; TeX output 1999.01.18:07370%8Р0+)3򞼍ǪfffftM tibJٙournalofGeodesymanuscriptNo. 8tM tir(willbeinsertedbytheeditor)34ffffffJtMG tibThedsolutionoftheKhornLichtensteinequationsofconformal0mapping:LՍtMff tibThedirectgenerationofellipsoidalGauKrugerconformalcoordinatesorthe ՍT4ransv"erseMercatorProjectionyE.Wz.Grafarend,R.Syffus-tM tirDepartment@ofGeodeticScience,StuttgzartUnifvpersityj=,Geschwister-Scholl-StrG.24D,D-70174Stuttgart,Germanpy Le-mail:@grafarend@gis.uni-stuttgzart.de捑Receifvped:@3June1997/Accepted:17November1997zIAbstract.ùTheXdifferentialequationswhichgenerateagen- <.eralSconformalmappingofatwfo-dimensional$tM tiiRiemannman-ifoldJofreferencegeneratestheconstraintsinquestion.FinallyY,the/detailedcomputationofthesolutionisgivٙenintermsofbivariatepolynomialsuptodeٙgree vewithcoef cientslistedinclosedform.!KeyRwfords.Kforn-Lichtenstein-equations !", cmsy10Conformalmap-pingEllipsoidofrevolution!ff^-L|1 Introductionx\ConvٙentionallyY,G conformalcoordinates/conformalchartsofthesurffaceoftheEarth,representedasanellipsoidofrev-olution,zthegeodeticreference gure,aregeneratedbyatwfo-step-procedure.PFirst,conformalcoordinates(isometriccoordinates,isothermalcoordinates)oftypeUMPf(Univٙer-sal8MercatorProjection,Example1)orUPS7(UnivٙersalPo-larStereographicProjection,Example2)arederivٙedfromgeodeticcoordinatessuchassurffacenormalellipsoidallon-gitude/ellipsoidal$:latitude.UMP$isclassi edasaconformalIMmappingonacircularcٙylinderwhileUPSreferstoacon- <.MformalmappingontoapolartangentialplanewithrespecttoMan(ellipsoidofrevolution(azimuthalmapping).Theconfor-MmalCcoordinatesoftypeUMP)orUPS,respectivٙelyY,arecon-Msequentlycompleٙxi ed,justdescribingthetwfo-dimensionalMRiemannMmanifoldofthetypeofellipsoidofrevolutionasaMone-dimensionalcompleٙxmanifold.NamelyY,thereal-valuedMconformalvcoordinatesK`y cmr10( b> cmmi10x;y[ٲ)oftypeUMPvorUPS,respec-MtivٙelyY,Haretransformedintothecomplex-valuedconformalMcoordinatez=D|x;+iy[ٹ.SecondlyY,theconformalcoordi-Mnates}o(x;y[ٲ);zoftypeUMP|orUPS,respectivٙelyY,areMtransformedZintoanothersetofconformalcoordinates,calledMGauKrÞzugerMorUTM,bymeansofholomorphicfunctionsMwD(zp);w:=>u5+iv2( msbm10C͹withrespecttocompleٙxalgebraMandcompleٙxanalysis.Indeedholomorphicfunctionsful ltheMd'AlembertEulereequations(Cauch3yRiemannequations)ofMconformalmappingasoutlinedbyGraffarend(1995),forin-Mstance.MThisDtwfo-step-procedurehasatleasttwobasicdisadvan-Mtages.|BOntheonehand,itisingeneraldif culttosetupaM rst bsetofconformalcoordinates.Fٙorinstance,duetoin-Mvolvٙed|dif cultiesthePhilosophicalFacultyoftheUniver-MsityofGÞzottingenGeorgiaAugustadated13June1857setMupthePreisaufg3abeto ndaconformalmappingoftheMtriaxial ellipsoid.BaseduponJacobi's3scontributiononellip-Mtic%coordinates(Jacobi,1839)thePreisschriftofScheringM(1857)hwfas nallycrowned,nevٙerthelessleavingthenumeri-McalproblemopenastohowtoconstructaconformalmapofMtheQtriaxialellipsoidoftypeUTM.FٙoranexcellentsurveyweMrefer:%toKlingenberg(1982),Schmehl(1927),and,recentlyY,MMÞzuller(1991).Thereisanotherdisadvantageofthetwfo-stepMprocedure..Theequivalence.betweentwfo-dimensionalreal-Mvaluedm Riemannmanifoldsandone-dimensionalcompleٙx-MvaluedmanifoldsholdsonlyforanalyticalRiemannmani-Mfolds. InGraffarend(1995)weg3avٙetwocountereٙxamplesofMsurffacesnfofrevolutionwhicharefromthedifferentiabilityclassMC^ O!cmsy71 0,butarenotanalytical.AccordinglyY,thetheoryofholo-MmorphicqfunctionsdoesnotapplyY.FinallyoneencountersgreatMdif cultiesvingeneralizingthetheoryofconformalmappingsMtoXhigher-dimensional(pseudo-)Riemannmanifolds.Onlyfor*0%8Ѝ20LEύ~1QFffUß&ٍ&ff ^vngeneral@coordinates 'kh(chart) Qparameters@of!a msbm9E-=Aacmr62;cmmi6A'Zcmr51 ;A'Ӱ2U&&ffffffUÎb IBff]";t;tff 5isometric@coordinates H0(conformal@coordinates) ,;of@typeMercator 卍#Projection@ofE-=2A'Ӱ1 ;A'Ӱ2 'ȍ /;complepxi cation\V;tffffff]"b ff]"E˖E˖ff [Ս5isometric@coordinates H0(conformal@coordinates) ,ߝof@typeGauKrcuger 卍of@E-=2A'Ӱ1 ;A'Ӱ2 ¿(T\ransvperse@Mercator]/Projection)\VE˖ffffff]"n~bK\ornLichtenstein H0 *pathN٠cfd굍O line10-+܄"$fe]?^w@^w@&^w@0^w@:^w@D^w@N^w@VW@VWRtM tibFigp.1.Changefromoneconformalcharttoanotherconformalchart L(c:c:Cha-Cha-Cha)aaccordingtoaproposalofGau(1822,1844); rst\uconformalcoordinates:MercatorProjection,secondconformalcoordinates:+OT\ransvperseMercatorProjection,ellipsoidofrefvolution 卑E-=2A'Ӱ1 ;A'Ӱ2Cevٙen-dimensional4(pseudo-)Riemannmanifoldsofanalytical <.type.canmultidimensionalcompleٙxanalysisbeestablished;weMeٙxperienceatotalffailureforodd-dimensional(pseudo-)Riemann7manifoldsastheٙyappearinthetheoryofrefraction,Newtonmechanics,plumblinecomputation,tolistjustafewconformallyatthree-dimensionalRiemannmanifolds.Thetheoryofconformalmappingtookquiteadifferentdi-rectionIwhenKforn(1914)aswellasLichtenstein(1911,1916)setuptheirgeneraldifferentialequationsfortwfo-dimensionalRiemann>manifoldswhichgoٙvern>conformalityY.TheٙyallowthestraightforwfardtransformationofellipsoidalcoordinatesoftypesurffacenormallongitudeLandlatitudeBb,intoconfor-mal-coordinatesoftypeGauKrÞzugerorUTM-(x;y[ٲ)withoutanٙyintermediateconformalcoordinatesystemoftypeUMPorS3UPS!Accordinglyourobjectivٙehereistheproofofthisstatement.Section2offersareviewoftheKfornLichtensteinequa-tionsofconformalmappingsubjecttotheinteٙgrabilitycon-ditionswhicharevٙectorialLaplaceBeltramiequationsonacurvٙedzsurfface,herewiththemetricoftheellipsoidofrevo-lution.޺T33wfoeٙxamples,namelyUMPޢandUPS,arechosentoshowүthatthemappingequationsx(L;Bq),y[ٲ(L;Bq)ful ltheKfornLichtensteinequationsaswellastheLaplaceBeltramiequations.Inaddition,wepresentintheAppendixafreshderivationeoftheKfornLichtensteinequationsofconformalmapping{fora(pseudo-)Riemannmanifoldofarbitrarydi-mensionAeٙxtendinginitialresultsforthree-dimensionalmani-foldsZofRiemanntypegivٙenbyZund(1987).ThestandardKfornLichtensteinequationsofaconformalmappingofatwfo-dimensionalMRiemannmanifoldcanbetakenfromstan-dardj=teٙxtbookslikfeBlaschkeandLeichtwei(1973)orHeitz(1988).Section 3aimsatasolutionofthepartialdifferentialequationsoftypeLaplaceBeltrami(second-order)aswellasXKfornLichtenstein( rst-order)inthefunctionspaceofbi-variateݒpolynomialsx(l2`;b),y[ٲ(l;b),lx:=LLٓRcmr70|s,ݒb:=Bq.B0|s.Thecoef cientconstraintsarecollectedinCorollaries1andLEύ~*~QFffUß&ٍ&ff ^vngeneral@coordinates 'kh(chart) Qparameters@ofE-=2A'Ӱ1 ;A'Ӱ2U&&ffffffUÎhܠIBffe7Y;t;tff  ѿisometric@coordinates H0؜(conformal@coordinates) ,of@typePolarStereogra- 卍phic@ProjectionofE-=2A'Ӱ1 ;A'Ӱ2 'ȍֿcomplepxi cationdj;tffffffe7Yhܟff]"E˖E˖ff [Ս5isometric@coordinates H0(conformal@coordinates) ,ߝof@typeGauKrcuger 卍of@E-=2A'Ӱ1 ;A'Ӱ2 ¿(T\ransvperse@Mercator]/Projection)\VE˖ffffff]"n~K\ornLichtenstein H0wpathT&cfd굍ά-_x܄"$fe?韂w@"韌w@,韖w@6韠w@@韪w@J韴w@T韾w@]>@]>RMFigp.2.Changefromoneconformalcharttoanotherconformalchart LM(c:c:Cha-Cha-Cha)accordingtoaproposalofKrcuger(1922); rstMconformal1coordinates:PolarStereographicProjection,secondcon-Mformalcoordinates:T\ransvperseMercatorProjection,ellipsoidofrefv- 卒Molution@E-=2A'Ӱ1 ;A'Ӱ2M2.Notethatthesolutionspaceisdifferentfromthatofsepara- <.MtionՀofvariablestypeknowntogeodesistsfromtheanalysisofMtheethree-dimensionalLaplaceBeltramiequationofthegrav-Mitationalpotential eld.FٙorarelateddiscussionseeGraffarendM(1995).MFinallyY,ASect4outlinestheconstraintstothegeneralsolu-MtionQoftheKfornLichtensteinequationssubjecttotheinteٙgra-MbilityconditionsoftypeLaplaceBeltramiequations,whichMleadsdirectlytotheconformalcoordinatesoftypeGauMKrÞzugerorUTM.Suchasolutionisgeneratedbytheequidis-MtantmappingofthemeridianofreferenceL0=(forUTMmuptoaMdilatation ffactor)astheproperconstraint(x(0;b)=0;y[ٲ(0;b)Mgivٙen).xYThehighlightisthetheoremwhichgivٙesthesolutionMofthepartialdifferentialequationsfortheconformalmappingMintermsofaconformalsetofbivariatepolynomials.Through-Mout,weusearight-handedcoordinatesystem,namelyxEast-Ming,߆y;_Northing.T33able4and5containthenon-vanishingMpolynomialcoef cientsinaclosedform."M2MTheequationsgofverٙningconformalmappingandtheirMfundamentalsolutionx\MW33e 6areconcernedherewithaconformalmappingofthebiax-Mial2ellipsoidE^2b 0ercmmi7A1 ;A2(ellipsoidofrevolution,spheroid,semi-Mmajor+axisA1|s,semi-minoraxisA2)embeddedinathree-MdimensionalEnEuclideanmanifoldE^3L=ٸfR^3|s;ij gwithstan-Mdard׻canonicalmetric[ij ],theKroneckferdeltaof1's3sintheMdiagonal,ofzerosintheoff-diagonal,namelybymeansofE^MX1/=<$eA1'cos5RB+cos9PL)ןwfeC u cmex10p fe9 18Er2sinƟ*2(BJ6;MX2/=<$ {A1'cos5RB+sinߵL)ןwfeC p fe9 18Er2sinƟ*2(BW (1)4MX3/=<$)׵A1|s(18E^2)sinnB)ןwfeI ߟ p fe9 18Er2sinƟ*2(B?0%8Ѝ׀30_"EintroducingqsurffacenormalellipsoidallongitudeLandsur- <.fface$normalellipsoidallatitudeBq.E^2xʲ:=h(A^2l1yA^2l2|s)=qA^2l1==1A^2l2|s=qA^2l1denotes<the rstrelativٙeeccentricitysquared.Ac-cordingto(L;Bq)2[[;)ܸ([=2;+=2)weeٙxcludefrom#thedomain(L;Bq)NorthandSouthPole.Thus(L;Bq)constitutesonlya rstchartofE^2bA1 ;A2`;aminimalatlasof AfE^2bA1 ;A2-ѹbasedxqontwfocharts,whichcoٙversxqallpointsoftheellipsoidmofrevolutionisgivٙenindetailbyGraffarend(1995).Conformalcoordinates(x;y[ٲ)(isometriccoordinates,isothermal\(coordinates)areconstructedfromthesurffacenor-mal!ellipsoidalcoordinates(L;Bq)assolutionsoftheKfornLichtensteinOequations(conformalchangefromonecharttoanotherchart,c:Cha-Cha-Cha)^!$9xL <-xBF^|=<$ѓ1Kwfe0 UNpUWfe(Y9UUEG8Fcr27^!$>SVFPE <->MԸGQFZg^aX؟^!$hµyL <-hyBui!^ɨ(2 )subjecttotheinteٙgrabilityconditions{ݍxLB%=xBW=L ^; yLB=yBW=LorOLB ^x:=^<$ ExmO \cmmi5B ,8FxmL Vwfe0 UNpUWfe(Y9UUEG8Fcr2=8^D BM+8^<$ ;ڵGxmL 8FxmB ȇwfe0 UNpUWfe(Y9UUEG8Fcr20ɰ(4)(orientationconservingconformeomorphism)ʍ[GM,N ](+:=^!$ 6E F <- FGV^$8M;N32f1;2gde nesthematrixofthemetricofthe rstfundamentalform <.of%E^2bA1 ;A2`.LB ^x=0;UPLBy,в=0,%respectivٙelyY,arecalledthevٙectorialLaplaceBeltramiequations.Aޣderivation޻oftheKfornLichtensteinequationsisgivٙeninEtheAppendix.HereweareinterestedinsomeeٙxamplesofmapprojectionsofconformaltypewhicharesolutionsoftherKfornLichtensteinequations[Eq.(2)]subjecttotheinte-grabilitypcondition[Eq.(4)]andtheconditionoforientationconservation[Eq.(4)].x\Example1.ǮUnivٙersalMercatorProjection(UMP){ݍx=~:A1|sLt$y=~:A1'ln'\ Ʋtan(.8^<$0ߵ0ߟwfe (֍4:9+<$lBlwfe9 (֍2 ^Wɟ^<$^718E>5sin/B^7wfe0ڟ (֍18+E>5sin/BK'^E۴Eb}=2i \!㒍The9wmatrixofthemetricoftheellipsoidofrevolutionE^2bA1 ;A2isrepresentedby'3[GM,N ]}=p^!$㎵EdF <-TFG5^"<=Vp2p6p6fip47E<$0A^2l1'cos^2ŵB zOwfe9 ݍ18Er2sinƟ*2(BjI0[$/0<$QȵA^2l1|s(18E^2)^2IuwfeFQ ݍ(18Er2sinƟ*2(Bq)r3V377fi5_"EMThemappingequationsoftheUMPimply MxLTݲ=A1|s; xB $=0;|qJyLTݲ=0; yB $=<${*A1|s(18E^2)KwfeZo ݍ(18Er2sinƟ*2(Bq)cosߵB6MKfornLichtensteinequationsʈMxLTݲ=5p 5fe! ˍE=G yB ; xB $=5p 5fe! ˍG=E"yLtorflqJyLTݲ=5p 5fe! ˍE=G"xB ; yB $=5p 5fe! ˍG=E!:xL M5pN5fe! ˍE=G:=cosOB<$18E^2sinƟ*2(Bwfe9 (֍18Er2I=UX)MyB $=<${*A1|s(18E^2)KwfeZo ݍ(18Er2sinƟ*2(Bq)cosߵBstq[:e:d:MinteٙgrabilityconditionssMLB ^x @=^\  r" fe [b<$33E33wfeT (֍ DG-xB \!A BJ+8\ #r#fe [b<$?wG33wfeT (֍EGxLt\!2 L:=0;"qLB ^y @=^\  r" fe [b<$33E33wfeT (֍ DG-yB \!@͜ BI舲+8\ #r#fe [b<$?wG33wfeT (֍EGyLt\!1M L:=0%b\M5pN5fe! ˍE=G soxB $=0;wM5pN5fe! ˍG=E soxLÌ=<$EA1|s(18E^2)Kwfed, ݍ(18Er2Hsin*2Bq)cosߵBj+;M(5p 5fe! ˍG=E"xLt)LÌ=0;M5pN5fe! ˍE=G soyB $=A1|s;M(5p 5fe! ˍE=G"yB )B $=0;M(5p 5fe! ˍG=E"yLt)=0q[:e:d: MorientationconservingconformeomorphismۨM M M M !$vxL*2xB <-ޡyL1yB CZ  CZ  CZ  CZ _Dz=xLtyB 8xB yLÌ=<${*A^2l1|s(1E^2)KwfeZo ݍ(1Er2sinƟ*2(Bq)cosߵBbd>0MduetoM[=20q:e:d:MThefzUMPesolutionoftheKfornLichtensteinequations <.Msubject]tothevٙectorialLaplaceBeltramiequationsasinte-Mgrability6conditionsandtheconditionoforientationconser-Mvation*isbasedontheconstraintofthefollowingtype.MapMtheequatorequidistantlyY,i.e.x(BG=0)=A1|s.x\MExample2.hUnivٙersalVePolarStereographicProjection(UPS);MxXi=<$~2A1Rwfe% UNpUWfeUU18Er2&0^<$.918E.9wfe.3 (֍18+EI!^P~۴Eb}=2-UPtanqŸ^<$iiwfe (֍4%|<$lBlwfe9 (֍2 ^CS^<$Iz18+E>5sin/BIzwfe0ڟ (֍18E>5sin/B{^ϟ۴Eb}=2W;coserLqyXi=<$~2A1Rwfe% UNpUWfeUU18Er2&0^<$.918E.9wfe.3 (֍18+EI!^P~۴Eb}=2UPtanqŸ^<$iiwfe (֍4%|<$lBlwfe9 (֍2 ^CS^<$Iz18+E>5sin/BIzwfe0ڟ (֍18E>5sin/B{^ϟ۴Eb}=2W;sinIL:0%8Ѝ40_"EThe9wmatrixofthemetricoftheellipsoidofrevolutionE^2bA1 ;A2 <.isrepresentedby(UA[GM,N ]}=p^!$㎵EdF <-TFG5^"<=Vp2p6p6fip47E<$0A^2l1'cos^2ŵB zOwfe9 ݍ18Er2sinƟ*2(BjI0[$/0<$QȵA^2l1|s(18E^2)^2IuwfeFQ ݍ(18Er2sinƟ*2(Bq)r3V377fi5(sThemappingequationsoftheUPSimply덍xL𳐲=zf(Bq)sinnL; xB $=f0Ȳ(B)cosߵL<.yL𳐲=zf(Bq)cosߵL; yB $=f0Ȳ(B)sinnLsubjectto9f(Bq)v:=<$2A1 ewfe% UNpUWfeUU18Er25C^<$=c18E=cwfe.3 (֍18+EX4ɟ^_=۴Eb}=2- 2UPtanqŸ^<$iiwfe (֍4%|<$lBlwfe9 (֍2 ^CS^<$Iz18+E>5sin/BIzwfe0ڟ (֍18E>5sin/B{^ϟ۴Eb}=2f0Ȳ(Bq)v:=<$2A1 ewfe% UNpUWfeUU18Er25C^<$=c18E=cwfe.3 (֍18+EX4ɟ^_=۴Eb}=2p^<$ݲ18E^2xPwfe9 ݍ18Er2sinƟ*2(BC^)fy 2UP^<$ ϡ18+E>5sin/B ϡwfe0ڟ (֍18E>5sin/B;㮟^A*̟۴Eb}=2򙖍Rktanbݟ^<$kkwfe (֍4u<$lBlwfe9 (֍2R^Rk fe>[ (֍mcos"|-Bg 2=<$k)(18E^2)wfeXN ݍcos7Bq(18Er2sinƟ*2(B)_tf(Bq)vKfornLichtensteinequations{fxL𳐲=z5pz5fe! ˍE=GLyB ; xB $=5p 5fe! ˍG=E"yLtorflyL𳐲=z5p 5fe! ˍE=G"xB ; yB $=5p 5fe! ˍG=E!:xL'ۍ5p5fe! ˍE=G:=cosOB<$18E^2sinƟ*2(Bwfe9 (֍18Er2?=UX),yB+=I<$18E^233wfeXN ݍcos7Bq(18Er2sinƟ*2(B)[+f(Bq)sinnL"+=If0Ȳ(Bq)sinnLq[:e:d:{yL+=I<$33cosAjBq(18E^2sinƟ*2(B)33wfeXN (֍T18Er2[+f0Ȳ(Bq)cosߵL1^+=If(Bq)cosߵLq[:e:d: <.!3AfundamentalsolutionfortheKorٙnLichtensteinequationsx\Fٙor-thebiaxialellipsoidE^2bA1 ;A2Nweshallconstructafunda-mental$CsolutionoftheellipsoidalKfornLichtensteinequa-tionss{forconformalmapping[Eq.(2)]subjecttothevٙecto-riallLaplaceBeltramiequations[Eq.(4)].Theconditionof_"EMorientationconservation[Eq.(4)]isautomaticallyful lled:~獍MxLTݲ=5p 5fe! ˍE=G yB ; xB $=5p 5fe! ˍG=E"yLtorflqJyLTݲ=5p 5fe! ˍE=G"xB ; yB $=5p 5fe! ˍG=E!:xL˞(5aG)Z荍M(5p 5fe! ˍG=E"xLt)L#ꑲ+.(5p 5fe! ˍE=G"xB )B $=0;qJ(5p 5fe! ˍG=E"yLt)L#ꑲ+.(5p 5fe! ˍE=G"yB )B $=0W (5b)MxLtyB 8xB yLÌ=5p 5fe! ˍG=E!:x2፴L5T+5p 85fe! ˍE=G! x2፴B $>0f(5c)M5pN5fe! ˍG=E :2Rǟ+W; 5p5fe! ˍE=G-C2Rǟ+!H(5d5)l <.MHereweareinterestedinalocalsolutionoftheellipsoidalMKfornLichtensteinequationsaroundapoint(L0|s;B0)չsuchMthatZL=L0!+l2`;BG=B0+bZhold.A5polynomialset-upoftheMlocallsolutionoftheellipsoidalKfornLichtensteinequationsMsubjecttotheellipsoidalvٙectorialLaplaceBeltramiequation~獍Myl=5p 5fe! ˍE=G"xbD; yb\=5p 5fe! ˍG=E!:xl˞(6aG)M(5p 5fe! ˍG=E"xlȲ)l9+(W(5p 5fe! ˍE=G"xbD)b\=0;flqJ(5p 5fe! ˍG=E"ylȲ)l9+(W(5p 5fe! ˍE=G"ybD)b\=0W (6b)lMisMx(l2`;b) 0=Nx0S+8x10xlk@+x01b+x20l2`2糲+x11l2`b+x02b2<.N+x30xl2`3糲+8x21l2`2ӵb+x12l2`b2S+x03b3S+OG(4)˞(7aG)qy[ٲ(l2`;b) 0=Ny0S+8y10xlk@+y01b+y20l2`2糲+y11l2`b+y02b2N+y30xl2`3糲+8y21l2`2ӵb+y12l2`b2S+y03b3S+OG(4)W (7b)lMorhMx(l2`;b)=Sf1 X n=0APnq~(l;b); y[ٲ(l;b)=Sf1 X n=0AQnq~(l;b)f(7c)$"XP0|s(l2`;b) 0 :=[x0XP1|s(l2`;b) 0 :=[x10xlk@+8x01b=X  + \=1x ȵl2` Bb 㔍XP2|s(l2`;b) 0 :=[x20xl2`2糲+8x11l2`b+x02b2C=X  + \=2x ȵl2` Bb !h . . .MPnq~(l2`;b) 0 :=-X [ + \=n5ix ȵl2` Bb !H(7d5)% XQ0|s(l2`;b) d:=:y0XQ1|s(l2`;b) d:=:y10xlk@+8y01b=X  + \=1y ȵl2` Bb 㔍XQ2|s(l2`;b) d:=:y20xl2`2糲+8y11l2`b+y02b2C=X  + \=2y ȵl2` Bb !h,d.,d.,d.MQnq~(l2`;b) d:= X : + \=n75y ȵl2` Bb ͹(7e)av0%8Ѝ׀50X0T,able@1.TGaylorepxpansionof5" cmmi9rAo cmr9(Br):=Mmzi cmex9p ŸMmaH/E2=G!v=cos~B(18 cmsy9E2-=28sinߟ-=2I0B)=(1E2-=2)=Lֳ̍N4(#cmex7P捳n=0P 71aH0n!rAǟ-=(n) '(B0*)b-=n=Lֳ̍N4(P捳n=0|rn7b-=n^7up@toorderthree)effr0 =+cosB0*(18E2-=28sinߟ-=2I0B0) aHYy۟)18E2j2 r1=+33sin9B0*(18+2E2-=23E2-=28sinߟ-=2I0B0)33 aHx}Y)._18E2j2ur2 =+33cos UB0*(18+2E2-=29E2-=28sinߟ-=2I0B0)33 aHyu))$ 2(18E2j2) r3=+sinB0*(18+20E2-=227E2-=28sinߟ-=2I0B0) aHU)-@|6(18E2j2) ff̎ȇ⍑T,able@2.TGaylorepxpansionofs(Br):=Mmp ŸMmaH/G=E!v=(18E2-=2)=[cos "B(1E2-=28sinߟ-=2I0B)]=s0JNЍ?Pˍn=0P 14aH0n!"s-=(n) O(B0*)b-=n=s0JNЍ?Pˍn=0sn7b-=n^7up@toorderthreeCi􍍍.<ffs0m=-ꍑ"̙18E2-=2ºaHYy۟jcos "B0*(18E2j28sinߟsK2I0B0)zUs1m=+(18E2-=2)sinpB0*(1+2E2-=23E2-=28sinߟ-=2I0B0) aHcjjcos)j2/fB0*(18E2j28sinߟsK2I0B0)j2D*s2m=+&s(18E2-=2) aHgџj2cos "j3sB0*(18E2j28sinߟsK2I0B0)j3lq/O18+2E2=2+sin i=2B0*(14E2=2+6E2=4)E2=28sinߟ=4I0B0*(2+13E2=2)+9E2=48sinߟ=6I0B0*s3m=+(18E2-=2)sinpB0 aHgџj6cos "j4sB0*(18E2j28sinߟsK2I0B0)j4lq/O58+4E2=2+24E2=4+sin i=2B0*(117E2=280E2=4+24E2=6)E2=28sinߟ=4I0B0*(591E2=2+68E2=4)ﮍ8E2=48sinߟ=6I0B0*(1765E2=2)27E2=68sinߟ=8B0*Љff0*IM subjecttotheT33ayloreٙxpansion2r5:=r fe [b<$33E33wfeT (֍ DGB:= Xcos(B<$18E^2sinƟ*2(Bwfe9 (֍18Er2U7B:= Xr0S+8r1|sb+r2b2S+r3b3S+OG(4)ɵ(8)l subjectto r0C:= α1r&fe@0!;rG^(0) (B0|s)-:(=rG(B0|s)S̍ r1C:= α1r&fe@1!;rG^(1) (B0|s)-:(=rG0V(B0|s)w鍍ȹ...Qˍrn8:= H1|&fe5 n!;rG^(n) (B0|s)-:(=<$ 1Kwfe>g (֍n(n81):::21CrG(n) (B0|s)1qs:=r fe [b<$?wG33wfeT (֍E(&=DrG1 ʩ=<$ 1Kwfe$p (֍cos7B<$+18E^2!wfe9 ݍ18Er2sinƟ*2(Bɵ(9)>(&=Ds0S+8s1|sb+s2b2S+s3b3S+OG(4)I􍍍(&=Dr䍐G10 g (֍n(n81):::21Cs(n) (B0|s)mBgivٙenEindetailbythecoef cientsinT33ables1and2.First,letus <.considerthevٙectorialLaplaceBeltramiequations[Eq.(6b)]4LB ^xj= 2(5p 5fe! ˍG=E"xlȲ)lJ+8(5p 5fe! ˍE=GxbD)b\=0flt$LB ^yj= 2(5p 5fe! ˍG=E"ylȲ)lJ+8(5p 5fe! ˍE=GybD)b\=0M Msxl `lp+8(rGxbD)b\=,sxl `lp+8rbDxb=$+rGxbb|=0ƞ(10aG)<.AGsyl `lp+8(rGybD)b\=-ߵsyl `lp+8rbDyb=$+rGybb|=0W (10b)œ,̵x(l2`;b)=#x0S+8x10xlk@+x01b+x20l2`2糲+x11l2`bW (11)#+x02xb2S+8x30l2`3糲+x21l2`2ӵb+x12l2`b2#+x03xb3S+8x40l2`4糲+x31l2`3ӵb+x22l2`2ӵb2#+x13xl2`b3S+8x04l4糲+8OG(5)<.xlȲ(l2`;b)=#x10 Ʋ+82x20xlk@+x11b+3x30l2`2糲+2x21l2`bJ>(12 )#+x12xb2S+84x40l2`3糲+3x31l2`2ӵb+2x22l2`b2#+x13xb3S+8OG(4)(19)(21),thenthefollowingmixٙedcoef cientrelationsMhold.%`; !Mn=1<-1y10w6=>Tr0|sx01 <-1y01w6=>Ts0|sx10W (26)OR!Mn=2<-02y20w6=>Tr0|sx11 <-1y11w6=>T2r0|sx02 Ƹ8r1x011y11w6=>T2s0|sx2002y02w6=>Ts0|sx11 Ʋ+8s1x10W (27)f<ۊ!Mn=3<-03y30w6=>Tr0|sx21 <-02y21w6=>T2r0|sx12 Ƹ8r1x111y12w6=>T3r0|sx03 Ƹ82r1x028r2x011y21w6=>T3s0|sx3002y12w6=>T2s0|sx21 Ʋ+82s1x2003y03w6=>Ts0|sx12 Ʋ+8s1x11+8s2x10W (28)e0%8Ѝ׀70Nn=4<-u4y40= r0|sx31 <-u3y31= 2r0|sx22 Ƹ8r1x21u2y22= 3r0|sx13 Ƹ82r1x128r2x11uy13= 4r0|sx04 Ƹ83r1x0382r2x028r3x01uy31= 4s0|sx40u2y22= 3s0|sx31 Ʋ+83s1x30u3y13= 2s0|sx22 Ʋ+82s1x21+82s2x20u4y04= s0|sx13 Ʋ+8s1x12+8s2x11+8s3x10ĵ(29)J-andingeneral% yl=LS1 X n=1|.n1 CX t i=0 (n8i)yni;il2`ni1bi"==61 X n=1fѴn1 X ti=0)*i #?kX t#jg=014(i8jk+1)rj6xni1;ijg+12 ;l2`ni1bi卍=rG(b)xb6cybc=>Ϸ1 ]X n=1 nn1  ̿X t |i=0(i8+1)yni1;i+1(l2`ni1bic=>Ϸ1 ]X n=1 nn1  ̿X t |i=0 i GDX tѴjg=0( (n8i)sj6xni;ijĵl2`ni1bic=s(b)xl*4TheconstraintstotheKorٙnLichtensteinequations <.whichgeneratetheGauKr"uger/UTMconformalmappingx\The'equidistantmappingofameridianofreferenceL0es-tablishes8theproperconstrainttotheKfornLichtensteinequa-tionsewhichleadstothestandardGauKr"ugerorUnivٙersalTfransvٙerseMercatorProjectionconformalmapping.ThearclengthofthecoordinatelineL0=const,namelythemerid-ian,betweenlatitudeB0sandBqiscomputedby$!y[ٲ(0;b)=cZi B@URB0S͟5pSΟ5feS ˍG(BqrU);յdBqm=Sf1 X n=1Ay0n mbnĦb(30[)$鍑aspbsoonaswesetupuniformlyconvٙergentT33aylorseriesoftype5p5fe# ˍG(Bq)<=<$A(18E^2)KwfeN2 ݍ(18Er2sinƟ*2(Bq)r3=2V'>=Sf1 X n=1<$O1twfeǷ (֍n!^G(n) (B0|s)bnĵ(31)$鍑andinteٙgratetermwise.T33able3isalistoftheresultingcoef- cientsby0n m,whichestablishtheset-upofthefollowingcon-straints.!F..._"EMReferenceszMBlaschkenW,,LeichtweiKm(1973)ElementareDiffferentialgeometrie. 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In:Conformalgeometryj=,KpulkarniRS,PinkallU(eds),Vuiefwepg,@BraunschweigWiesbadenMLichtensteinyL(1911)BefweisdesSatzes,dajedeshinreichendkleine imwesentlichenstetiggekrcummte,singularitatenfreieFlachenstcuckaaufeinenT^eileinerEbenezusammenhangendundindenkleinstenT^eilenlݞahnlichabgebildetwerdenkann.AbhKconiglD PreussAkadWissBerlin,PhzysMathKD AnhangAbhVI:143MLichtensteinL(1916)ZurTheoriederkonformenAbbildung.BullInt@AcadSciCracopvieSerA:192217MLiouville fJ ^(1850)Extensionaucasdestroisdimensionsdelaques-tiondutracegeographique.NoteVI,byG.Monge:applicationdeRl'analyse2alageometrie,cinquieme2editionrefvuecorrigeeparM.@Liouville,Bachelier,PparisMMculler1 cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5O line10u cmex10