(Email: qddqinfen@cgs.gov.cn) http://www.posc.org Coordinate Conversions and Transformation including Formulas part 2 Formulas for Coordinate Operations other than Map Projections EPSG 1995 2004 http://www.globalsecurity.org/military/library/policy/army/fm/6-2/appe2.htm#tabe_22 GPS GPS WGS84 WGS84 XYZ XYZ WGS84 GPS dxdydz
Molodenski Helmert7 EPSG - Bursa-Wolf - Molodenski-Badekas 7 1927 Clarke 1966 1983 GRS 1980 Clarke 1966 Meades Ranch Molodenski Helmert EPSG NAD83 US Coast & Geodetic SurveyNADCON EPSG NAD27 NAD83 ED87 1987 ED50 1950 Statens Kartwerk 15 4 From To ϕ λ XYZ Z X Y 90
a b 1/f X = ( ν ϕ λ Y = ( ν ϕsin λ Z = 2 (( ) ν ϕ ν ϕ ν = a/(1 e sin ϕ) 2 2 0.5 ϕ λ h e e = ( a b )/ a = 2f f 2 2 2 2 2 N 100 +60 XYZ ϕ = tg [( Z + eν sin ϕ)/( X + Y ) 1 λ = tg ( Y / X ) 1 2 2 2 0.5 h= X secλsecϕ ν λ 3.2 AA > 0 X = X A new old EPSG 1999 X = X + A t s Xs Xt A
EPSG Xt = Xs + A X = [( X * U ) + ( A* U )]*( m / U ) t s s A t Xt = Xs = A m m=1 m = 1 UsUt U A EPSG dxdydz X = X + dx t t t Y = Y + dy s Z = Z + dz s s 2.1 WGS84 GPS ϕ = 53 48'33.82''N s λ = 207'46.38''E s h = 73.0m s ED50 International 1924 WGS84 ED50 dx = +84.87m dy = +96.49mdZ = +116.95m WGS84 X Y s Z s s = 3771793.97m = 140253.34m = 5124304.35m ED50 X t t t = 3771793.97+ 84.87 = 3771878.84m Y = 140253.34+ 96.49 = 140349.83m Z = 5124304.35+ 116.95= 5124421.30m
2 ED50 ϕ = 53 48'36.565''N t λ = 207'51.477''E t h = 28.02m t International 1924 Molodenski Molodenski ϕ = ϕ + dϕ t λ = λ + dλ t h = h + dh t s s s dϕ'' = ( dx.sin ϕ.cos λ dy.sin ϕ.sin λ+ dz.cos ϕ + [ adf. + fda. ].sin2 ϕ)/( ρ.sin1'') dλ'' = ( dx.sin λ+ dy.cos λ)/(.cos ν ϕ.sin1'') 2 dh = dx.cos ϕ.cos λ + dy.cos ϕ.sin λ + dz.sin ϕ + ( adf. + fda. ).sin ϕ da dxdydz ρ ϕ ρ = a(1 e )/(1 e sin ϕ) ν ϕ ν = a/(1 e sin ϕ) 2 2 2 3/2 2 2 1/2 da da = a a df df = f f = 1/(1/ f) 1/(1/ f ) dϕ λ ϕ λ t s t s t s WGS84 GPS ϕ = 53 48'33.82''N s λ = 207'46.38''E s h = 73.0m s ED50 International 1924 WGS84 ED50 dx = +84.87m dy = +96.49m dz = +116.95m
WGS 1984 a = 6378137.0 1/f = 298.2572236 International 1924 a = 6378388.0 1/f = 297.0 da = 6378137 6378388 = 251 df = 0.003352811 0.003367003 = 1.41927E-05 dϕ = 2.545'' dλ = 5.097'' dh= 44.98m ED50 International 1924 ϕ = 53 48'36.565'' N t λ = 207'51.477'' E t = 28.02m t Helmert 7 7 XT 1 R + R Z Y XS dx Y T M * RZ 1 RX * Y S dy = + + Z T RY RX 1 + Z S dz XsYsZs XtYtZt EPSG dxdydz RXRYRZ M M = (1 + ds*10-6 ) ds ppm
2.1 WGS72 WGS84 dx = 0.000 m dy = 0.000 m dz = +4.5 m RX = 0.000 RY = 0.000 RZ = +0.554 ds = +0.219 ppm WGS72 ϕ = 55 00'00''N S λ = 400'00''E S h = 0m S X Y Z S S S = 3657660.66m = 255768.55m = 5201382.11m 7 X Y T Z T T = 3657660.78m = 255778.43m = 5201387.75m WGS84 ϕ = 55 00'00.090''N T λ = 400'00.554''E T h =+ 3.22m T EPSG XT 1 + R R Z Y X S dx Y T M * RZ 1 RX * Y S dy = + + Z T RY RX 1 + Z S dz dxdydz
RXRYRZ M M = (1 + ds*10-6 ) ds ppm dx = 0.000 m dy = 0.000 m dz = +4.5 m RX = 0.000 RY = 0.000 RZ = 0.554 ds = +0.219 ppm ISO 19111 Molodenski-Badekas XT 1 + R R Z Y XS XP X P dx Y T M * RZ 1 RX * YS Y P Y P dy = + + + Z T RY RX 1 + ZS Z P Z P dz dxdydz RXRYRZ XPYPZP
M M = (1 + ds*10-6 ) ds ppm Molodenski-Badekas 6*10 1 6*10 2 6.3*10 6 XY 2 3 1010 X S0 Y S0 X T0 Y T0
X Y S S X Y S0 S0 X Y T T X Y T0 T0 U = m ( X X ) S S S0 V = m ( Y Y ) S S S0 X S Y S X S0 Y S0 m S UV AnBn dxdy m T m dx = A + A U + A V + A U + A U V + A V T 2 2 0 1 2 3 3 5 + A U + A U V+ A U V + A V 3 2 2 3 6 7 8 9 + A U + A U V+ A U V + A U V + A V 4 3 2 2 3 4 10 11 12 13 14 + A U + A U V + A U V + A U V + A U V + A V 5 4 3 2 2 3 4 5 15 16 17 18 19 20 + A U + A U V + A U V + A U V + A U V + A U V + A V 6 5 4 2 3 3 21 22 23 24 25 +... + A V 104 13 2 4 5 6 26 27 m dy= B + B U + B V + B U + B U V + B V T 2 2 0 1 2 3 3 5 + B U + B U V+ B U V + B V 3 2 2 3 6 7 8 9 + B U + B U V+ B U V + B U V + B V 4 3 2 2 3 4 10 11 12 13 14 + B U + B U V + B U V + B U V + B U V + B V 5 4 3 2 2 3 4 5 15 16 17 18 19 20 + B U + B U V + B U V + B U V + B U V + B U V + B V 6 5 4 2 3 3 21 22 23 24 25 +... + B V 104 13 2 4 5 6 26 27 dxdy EPSG Aumvn Bumvn m U n V A17 Au3v2
( X X ) = ( X X ) + dx T T0 S S0 ( Y Y ) = ( Y Y ) + dy T T0 S S0 X = X X + X + dx T S S0 T0 Y = Y Y + Y + dy T S S0 T0 X T Y T X S Y S X S0 Y S0 X T0 Y T0 dxdy 6 EPSG 9648 TM75 ETRS89 X0X 0 = ϕ 0 = 5330'00.000'' N =+ 53.5 S Y0Y 0 = λ 0 = 742'00.000'' W = 7.7 S S S X0X 0 = ϕ 0 = 5330'00.000'' N =+ 53.5 T Y0Y 0 = λ 0 = 742'00.000'' W = 7.7 T T m S = 0.1 m T = 3600 T A0 = 0.763 A1 = 4.487 A24 =265.898 A27 = 0 B0 = 2.810 B1 = 0.341 B24 = 853.950 B27 = 0 TM75 X Y ϕ λ TM75 TM75 S S0 TM75 S0 S S0 TM75 S0 = X = 55 00'00'' N =+ 55.000 S = Y = 630'00'' W = 6.500 S X = ϕ ϕ = 55.0 53.5 = 1.5 Y = λ λ = 6.6 ( 7.7) = 1.2 U = m ( X X ) = m ( ϕ ϕ ) = 0.1*(1.5) = 0.15 S S S0 S TM75 S0 V = m ( Y Y ) = m ( λ λ ) = 0.1*(1.2) = 0.12 S S S0 S TM75 S0 3 3 dx = ( A0 + A1 U+... + A24 U V )/ K TCD dx 3 3 = {0.763 + ( 4.487*0.15) +... + ( 265.898*0.15 *0.12 )}/3600= 0.00003046
3 3 dy = ( B0 + B1 U+... + B24 U V )/ K TCD dy 3 3 = { 2.81 + ( 0.341*0.12) +... + ( 853.95*0.12 *0.12 )}/3600= 0.00094799 ϕ λ = X = X + dx = 55.0+ 0.00003046= 55 00'10.9656'' N ETRS89 T S = Y = Y + dy = 6.6 0.00094799= 630'03.4128'' W ETRS89 T S 1 2 EPSG n 1 2 3 4 dxdy 1 X SO = X T0 = X 0 Y SO = Y T0 = Y 0 m S = m T = m 3 123 dxdy dxdy dxdy ED50 ED87 1 2 ED50 ED87 2 10-6 3 55N0E4 dx dy 10-6
10 6 EPSG 4 EPSG 9651 ED50 ED87 X Y = ϕ = 55 00'00.000'' N =+ 55 0 0 = λ = 000'00.000'' E =+ 0 0 0 m = 1.0 A0 = 5.56098E-06 A1 = 1.55391E-06 A14 =4.01383E-09 B0 = +1.48944E-05 B1 = +2.68191E-05 B14 = +7.62236E-09 ED50 ϕ λ ED50 ED50 = X = 52 30'30'' N =+ 52.508333333 S = Y = 2 E =+ 2 S U = m ( X X ) = m ( ϕ ϕ ) = 1.0*(52.508333333 55.0) = 2.491666667 S 0 ED50 0 V = m ( Y Y ) = m ( λ λ ) = 1.0*(2.0 0.0) = 2 S 0 ED 50 0 4 dx = ( A0 + A1 U+... + A14 V )/ K CD = [ 5.56098E 06 + ( 1.55391E 06* 2.491666667) +... + ( 4.01383E 09*2.0 4)]/1.0 = 3.12958E 06 4 dy = ( B0 + B1 U+... + B14 V )/ K CD = [ + 1.48944E 05 + (2.68191E 05* 2.491666667) +... + (7.62236E 09*2.0 4)]/1.0 =+ 9.80126E 06 ϕ λ = X = X + dx = 52.508333333 3.12958E 06= 52 30'29.9887'' N ED87 T S = Y = Y + dy = 200'00.0353'' E ED87 T S
ED87 ED50 1 4 A0A14B0B14 X Y = ϕ = 55 00'00.000'' N =+ 55 0 0 = λ = 000'00.000'' E =+ 0 0 0 m = 1.0 A0 = +5.56098E-06 A1 = 1.55391E-06 A14 =4.01383E-09 B0 = 1.48944E-05 B1 = 2.68191E-05 B14 = 7.62236E-09 ED50 ϕ λ ED87 ED87 = X = 52 30'29.9887'' N =+ 52.5083301944 S = Y = 200'00.0353'' E =+ 2.0000098055 S U = 1.0*(52.5083301944 55.0) = 2.4916698056 V = 1.0*(2.0000098055 0.0) = 2.0000098055 4 dx = ( A0 + A1 U+... + A14 V )/ K CD = [ + 5.56098E 06 + (1.55391E 06* 2.4916698056) +... + (4.01383E 09*2.0000098055 4)]/1.0 = 3.12957E 06 4 dy = ( B0 + B1 U+... + B14 V )/ K CD = [ 1.48944E 05 + ( 2.68191E 05* 2.4916698056) +... + ( 7.62236E 09*2.0000098055 4)]/1 = 9.80124E 06 ϕ λ = X = X + dx = 52.5083301944+ 3.12957E 06= 52 30'30.000'' N ED50 T S = Y = Y + dy = 200'00.000'' E ED50 T S
ABUV 3 m ( dx + idy ) = ( A + i A )( U + iv ) + ( A + i A )( U + iv ) T 1 2 3 4 + ( A + i A )( U + iv ) + ( A + i A )( U + iv ) 3 4 5 6 7 8 U = m ( X X ) S S S0 V = m ( Y Y ) S S S0 m S m T 2 4 Amersfoort / RD ED50 / UTM U V 2 2 U V m dx 2 T + A1 A2 + A3 A4 + A5 A UV 6 + A7 A8 * 3 2 m U 3UV T dy = A2 A1 A4 A3 A6 A5 A8 A + + + + + + + + 7 2 3 3UV V 4 2 2 4 U 6UV + V 3 3 4UV 4UV X = X X + X + dx T S S0 T0 Y = Y Y + Y + dy T S S0 T0 X T Y T X S Y S X S0 Y S0 X T0 Y T0 0 A0 B0
4 EPSG 9653 Amersfoort / RD New ED50 / UTM zone 31N X S0 155000.000 Y S0 463000.000 X T0 663395.607 Y T0 5781194.380 m S 10-5 m T 1.0 A1 A1-51.681 A2 A2 +3290.525 A3 A3 +20.172 A4 A4 +1.133 A5 A5 +2.075 A6 A6 +0.251 A7 A7 +0.075 A8 A8-0.012 X Amersfoort / RD = X S = 200000.00 m Y Amersfoort / RD = Y S = 500000.00 m U = m X X = = 5 S ( S S0) (200000 155000) 10 0.45 V = m Y Y = = 5 S ( S S0) (500000 463000) 10 0.37 dx =1240.050/ 1.0 dy =1468.748/ 1.0 X ED50 / UTM 31 = X T = X S X S0 + X T0 + dx = 707155.557 m Y ED50 / UTM 31 = Y T =Y S Y S0 + Y T0 + dy = 200000 155000 + 663395.607 +1240.050 = 500000 463000 + 5781194.380 + 1468.748 = 5819663.128 m
EPSG 9617 Madrid 1870 Struve 1860 Madrid El Servicio Geografico del Ej rcito Madrid 1870 ED50 1950 Madrid 1870 ED50 dϕ( ) = A + ( A * ϕ ) + ( A * λ ) + ( A * H ) 0 1 S 2 S 3 S dλ( ) = B + B + ( B * ϕ ) + ( B * λ ) + ( B * H ) 00 0 1 S 2 S 3 S ϕ s λ s Madrid 1870 Hs B 00 Madrid Greenwich B 00 ϕ = ϕ + dϕ ED50 M1870( M) λ = λ + dλ ED50 M1870( M) Madrid 1870 ϕ = 42 38'52.77'' N =+ 42.647992 S λ = 339'34.57'' EMadrid ( ) =+ 3.659603 ( Madrid) S H S = 0 m A0 = 11.328779 B00 = 13276.58 A1 =0.1674 B0 = 2.5079425 A2 =0.03852 B1 = 0.8352 A3 = 0.0000379 B2 = 0.00864 dϕ =+ 4.05'' ϕ ED 50 ED50 B3 = 0.0000038 = 42 38'52.77'' N + 4.05" = 42 38'56.82'' N d λ = 13270.54'' = 341'10.54'' λ = 339'34.57'' E 341'10.54'' = 001'35.97''( Greenwich)
EPSG EPSG 9624 CAD GIS rubber sheeting XT A0 A X 1 A2 S * Y = B + B B Y T 0 1 2 S X = A + A X + A Y T 0 1 S 2 S Y = B + B X + B Y T 0 1 S 2 S X T Y T P X S Y S P 4.1 D= A B A B 1 2 2 1 A ' = ( A B B A )/ D 0 2 0 2 0 B ' = ( B A A B )/ D 0 1 0 1 0 A ' =+ B / D 1 2 A ' = A / D 2 2 B ' = B / D 1 1 B ' =+ A / D 2 1
EPSG 9623 9 X = X + Y sinθ + X cosθ TP T0 SP Y SP X Y = Y + Y cosθ X sinθ TP T0 SP Y SP X ds X ds Y k k 0
k k X TP Y TP X SP Y SP P X = X + X k ds cosθ + Y k ds sinθ T T0 S X X S Y Y Y = Y X k ds sinθ + Y k ds cosθ T T0 S X X S Y Y XT XT0 cosθx sinθy k ds X 0 XS = + * * Y Y sinθ cosθ 0 k ds Y T T0 X Y Y S X TO Y TO ds X ds Y k X Y 4.2.1 A = X A 0 T 0 1 A B B B 2 0 T 0 1 2 = k ds X = k ds = Y Y = k ds = k ds Y *cosθ *sinθ X Y *sinθ *cosθ X Y X 1 2 XS 1 1/ ds 0 cos sin 0 * X θ T T * Y θy X X = * YS k D 0 1/ ds Y sinθx cosθ X YT YT0 D = cos( θ θ ) X Y X = [( X X )cos θ ( Y Y )sin θ ]/[ k ds cos( θ θ )] S T T0 Y T T0 Y X X Y Y = [( X X )sin θ + ( Y Y )cos θ ]/[ k ds cos( θ θ )] S T T0 X T T0 X Y X Y
EPSG 9622 X = Y = XT XT0 cosθ sinθ k dsx 0 XS = + * * Y Y sinθ cosθ 0 k ds Y T T0 Y S X = X + X k ds cosθ + Y k ds sinθ T T0 S X S Y Y = Y X k ds sinθ + Y k ds cosθ T T0 S X S Y X TO Y TO ds X ds Y k Y Y X Y
EPSG 9621 ds X = ds Y = ds 10 X = X + Y sinθ+ X cosθ TP T0 SP SP Y = Y + Y cosθ X sinθ TP T0 SP SP X TP Y TP X SP Y SP P X = X + X ds cosθ + Y ds sinθ T T0 S S Y = Y X ds sinθ + Y ds cosθ T T0 S S XT XT0 cosθ sinθ XS = + (1 + ds)* * Y Y sinθ cosθ Y T T0 S X TO Y TO ds
Y Y A1 = B2 A2 = B1 ds 1 1/1+dS1-dS + X TO Y TO 1 2 3 1 Astra Minas Grid Campo Inchauspe / Argentina 2 Astra Minas Grid X Y Campo Inchauspe / Argentina 2 X Y
Astra Minas Grid XY Campo Inchauspe / Argentina 2 YX X S = Astra Minas X Y S = Astra Minas Y X T = Campo Inchauspe / Argentina 2 Y Y T = Campo Inchauspe / Argentina 2 X X T0 = 2610200.48 m Y T0 = 4905282.73 m = 2710530 k = 0 1 + k= 1.0 Astra Minas X= 10000 m Y= 50000 m X S = Astra Minas X = 10000 Y S = Astra Minas Y = 50000-2 Y= X = X 0 + X ds cosθ + Y ds sinθ T T S S = 2610200.48 +50000*1.0*cos271.0916667+10000*1.0*sin271.0916667 = 2601154.90 m - 2 X= Y = Y 0 X ds sinθ + Y ds cosθ T T S S = 4905282.73 50000*1.0*sin271.0916667+10000*1.0*cos271.0916667 = 4955464.17 m 2 3 I 25 I 12.5 WGS84 /UTM Zone 31N E = 456781.0N = 5836723.0 0.99984 I J 11020 +20 X T0 = 456781.0 m Y T0 = 5836723.0 m ds X = 25 ds Y = 12.5 k = 0.99984 = +20
I= 300 J = 247 X = = X + X k ds cosθ + Y k ds sinθ = 464855.62m T T0 S X S Y Y = = Y X k ds sinθ + Y k ds cosθ = 5837055.90m T T0 S X S Y X = [( X X )cos θ ) ( Y Y )sin θ )]/[ k ds cos( θ θ )] = 230 S T T0 Y T T0 Y X X Y Y = [( X X )sin θ ) + ( Y Y )cos θ )]/[ k ds cos( θ θ )] = 162 S T T0 X T T0 X X X Y EPSG 1995200
http://www.globalsecurity.org/military/library/policy/army/fm/6-2/appe2.htm#tabe_22 semimajor axis a semiminor axis b flattening (1/f) b = a (1 - f)
WGS84 a WGS84 f X WGS84 10 7 XYZ WGS84 Airy
Modified Airy
Australian National
Bessel 1841
Bessel 1841Namibia
Clarke 1866
Clarke 1880
Everest 1830
Everest 1948
Modified Everest 1948
Modified Everest 1956
EverestPakistan
Modified Fischer 1960
Geodetic ReferenceSystem 1980
Geodetic ReferenceSystem 1980 China
Helmert 1906
Hough 1960
Indonesian 1974
International 1924
Krassovsky 1940 Krassovsky International 2030
Soviet Geodetic System 1985
South American 1969
２３ Geodetic ReferenceSystem 1972 椭球体基准面转换参数 WGS72 转换到 WGS84 转换公式如下 该公式计算精度比简化莫洛金斯基公式高 所 以不需要通过三参数平移