9601,Longitude rotation,,This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.,Longitude rotation,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Lon2 = Lon1 + longitude_rotation.,,1999-11-12 00:00:00,,EPSG,,99.79
9602,Geodetic/geocentric conversions,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"Latitude, P, and Longitude, L, in terms of Geographic Coordinate System (GCS) A may \
be expressed in terms of a geocentric (earth centred) cartesian coordinate system X, Y, Z \
with the Z axis corresponding with the Polar axis positive northwards, the X axis through \
the intersection of the Greenwich meridian and equator, and the Y axis through the \
intersection of the equator with longitude 90 degrees E. If the GCS's prime meridian is not \
Greewich, longitudes must first be converted to their Greenwich equivalent. If the earth's \
spheroidal semi major axis is a, semi minor axis b, and inverse flattening 1/f, then\
\
XA= (nu + hA) cos P cos L\
YA= (nu + hA) cos P sin L\
ZA= ((1 - e^2) nu + hA) sin P\
\
where nu is the prime vertical radius of curvature at latitude P and is equal to \
nu = a /(1 - e^2*sin^2(P))^0.5,\
P and L are respectively the latitude and longitude (related to Greenwich) of the \
point \
h is height above the ellipsoid, (topographic height plus geoidal height), and\
e is the eccentricity of the ellipsoid where e^2 = (a^2 -b^2)/a^2 = 2f -f^2\
Cartesian coordinates in geocentric coordinate system B may be used \
to derive geographical coordinates in terms of geographic coordinate system B by:\
P = arctan (ZB + e^2* nu*sin P) / (XB^2 + YB^2)^0.5 by iteration\
L = arctan YB/XB\
hB = XB sec L sec P - nu\
\
where LB is relative to Greenwich. If the geographic system has a non Greenwich prime \
meridian, the Greenwich value of the local prime meridian should be applied to longitude.\
\
(Note that h is the height above the ellipsoid. This is the height value which is \
delivered by Transit and GPS satellite observations but is not the topographic \
height value which is normally used for national mapping and levelling operations. \
The topographic height is usually the height above mean sea level or an alternative \
level reference for the country. If one starts with a topographic height, it will be \
necessary to convert it to an ellipsoid height before using the above transformation \
formulas. h = N + H, where N is the geoid height above the ellipsoid at the point \
and is sometimes negative, and H is the height of the point above the geoid. The \
height above the geoid is often taken to be that above mean sea level, perhaps with \
a constant correction applied. Geoid heights of points above the nationally used \
ellipsoid may not be readily available. For the WGS84 ellipsoid the value of N, \
representing the height of the geoid relative to the ellipsoid, can vary between \
values of -100m in the Sri Lanka area to +60m in the North Atlantic.)","Consider a North Sea point with coordinates derived by GPS satellite in the WGS84 geographical coordinate system with coordinates of:\
\
latitude 53 deg 48 min 33.82 sec N, \
longitude 02 deg 07 min 46.38 sec E, \
and ellipsoidal height 73.0m, \
\
whose coordinates are required in terms of the ED50 geographical coordinate system which takes the International 1924 ellipsoid. The three parameter datum shift from WGS84 to ED50 for this North Sea area is given as dX = +84.87m, dY = +96.49m, dZ = +116.95m. \
\
The WGS84 geographical coordinates convert to the following geocentric values using the above formulas for X, Y, Z:\
\
XA = 3771 793.97m\
YA = 140 253.34m\
ZA = 5124 304.35m\
\
Applying the quoted datum shifts to these, we obtain new geocentric values now related to ED50:\
\
XB = 3771 878.84m\
YB = 140 349.83m\
ZB = 5124 421.30m\
\
These convert to ED50 values on the International 1924 ellipsoid as:\
latitude 53 deg 48 min 36.565 sec N, \
longitude 02 deg 07 min 51.477 sec E, \
and ellipsoidal height 28.02 m, \
\
Note that the derived height is referred to the International 1924 ellipsoidal surface and will need a further correction for the height of the geoid at this point in order to relate it to Mean Sea Level.",1996-09-18 00:00:00,"\"Transformation from spatial to geographical coordinates\"; B. R. Bowring; Survey Review number 181; July 1976.",EPSG,,97.29
9603,Geocentric translations,,,X-axis translation,Y-axis translation,Z-axis translation,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"If we may assume that the minor axes of the ellipsoids are parallel, then shifts dX, dY, dZ \
in the sense from datum A to datum B may then be applied as\
\
XB = XA + dX \
YB = YA + dY\
ZB = ZA + dZ","Given a three parameter datum shift from WGS84 to ED50 for this North Sea area is given as \
dX = +84.87m, dY = +96.49m, dZ = +116.95m. \
\
The WGS84 geographical coordinates convert to the following GS84 geocentric values using \
the above formulas for X, Y, Z:\
\
XA = 3771 793.97m\
YA = 140 253.34m\
ZA = 5124 304.35m\
\
Applying the given datum shifts to these, we obtain new geocentric values now related \
9605,Abridged Molodenski,,,X-axis translation,Y-axis translation,Z-axis translation,Semi-major axis length difference,Flattening difference,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"As an alternative to the computation of the new latitude, longitude and height above ellipsoid in discrete steps through geocentric coordinates, the changes in these coordinates may be derived directly by formulas derived by Molodenski. Abridged versions of these formulas, which are quite satisfactory for three parameter transformations, are as follows:\
where the dX, dY and dZ terms are as before, and rho and nu are the meridian and prime vertical radii of curvature at the given latitude (lat) on the first ellipsoid (see section 1.4), Da is the difference in the semi-major axes (a1 - a2) of the first and second ellipsoids and Df is the difference in the flattening of the two ellipsoids.\
\
The formulas for Dlat and Dlon indicate changes in latitude and longitude in arc-seconds.",,1999-04-22 00:00:00,,EPSG,,99.01
9606,Position Vector 7-param. transformation,,,X-axis translation,Y-axis translation,Z-axis translation,X-axis rotation,Y-axis rotation,Z-axis rotation,Scale difference,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"Transformation of coordinates from one geographic coordinate system into another (also known as a \"datum transformation\") is usually carried out as an implicit concatenation of three transformations:\
[geographical to geocentric >> geocentric to geocentric >> geocentric to geographic]\
\
The middle part of the concatenated transformation, from geocentric to geocentric, is usually described as a simplified 7-parameter Helmert transformation, expressed in matrix form with 7 parameters, in what is known as the \"Bursa-Wolf\" formula:\
\
(XÆ) ( 1 -Rz +Ry) (X) (dX)\
(YÆ) = M * ( +Rz 1 -Rx) * (Y) + (dY)\
(ZÆ) ( -Ry +Rx 1 ) (Z) (dZ)\
\
The parameters are commonly referred to defining the datum transformation \"from Datum 'A' to Datum 'B'\", whereby (X, Y, Z) are the geocentric coordinates of the point on Datum æAÆ and (XÆ, YÆ, ZÆ) are the geocentric coordinates of the point on Datum æBÆ. However, that does not define the parameters uniquely; neither is the definition of the parameters implied in the formula, as is often believed. However, the following definition, which is consistent witth the \"Position Vector Transformation\" convention, is common E&P survey practice: \
\
(dX, dY, dZ) :Translation vector, to be added to the point's position vector in coordinate system 'A' in order to transform from system 'A' to system 'B'; also: the coordinates of the origin of system 'A' in the 'B' frame.\
\
(Rx, Ry, Rz) :Rotations to be applied to the point's vector. The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate system in the positive direction of that axis. E.g. a positive rotation about the Z-axis only from system 'A' to system 'B' will result in a larger longitude value for the point in system 'B'.\
\
M : The scale correction to be made to the position vector in coordinate system 'A' in order to obtain the correct scale of coordinate system 'B'. M = (1+S*10 6), whereby S is the scale correction expressed in parts per million. \
\
<<<<<This text continues in the description of the Coordinate Frame Rotation formula>>>>>","Input point: \
Coordinate system: WGS72 (geographic 3D)\
Latitude = 55 deg 00 min 00 sec \
Longitude = 4 deg 00 min 00 sec \
Ellipsoidal height = 0 m\
This transforms to cartesian geocentric coords:\
X = 3 657 660.66 (m) \
Y = 255 768.55 (m)\
Z = 5 201 382.11 (m)\
\
Transformation parameters WGS72 to WGS84:\
dX (m) = 0.000 \
dY (m) = 0.000 \
dZ (m) = +4.5\
RX (\") = 0.000 \
RY (\") = 0.000\
RZ (\") = +0.554\
Scale (ppm) = +0.219\
\
Application of the 7 parameter Position Vector Transformation results in WGS 84 coordinates of:\
9607,Coordinate Frame rotation,,,X-axis translation,Y-axis translation,Z-axis translation,X-axis rotation,Y-axis rotation,Z-axis rotation,Scale difference,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"<<<<<This text is continued from the description of the Position Vector Transformation formula>>>>>\
\
Although being common practice in particularly the European E&P industry Position Vector Transformation sign convention is not universally accepted. A variation on this formula is also used, particularly in the USA E&P industry. That formula is based on the same definition of translation and scale parameters, but a different definition of the rotation parameters. The associated convention is known as the \"Coordinate Frame Rotation\" convention. \
The formula is:\
\
(XÆ) ( 1 +Rz -Ry) (X) (dX)\
(YÆ) = M * ( -Rz 1 +Rx) * (Y) + (dY)\
(ZÆ) ( +Ry -Rx 1 ) (Z) (dZ)\
\
and the parameters are defined as:\
\
(dX, dY, dZ) : Translation vector, to be added to the point's position vector in coordinate system 'A' in order to transform from system 'A' to system 'B'; also: the coordinates of the origin of system 'A' in the 'B' frame.\
\
(Rx, Ry, Rz) : Rotations to be applied to the coordinate frame. The sign convention is such that a positive rotation of the frame about an axis is defined as a clockwise rotation of the coordinate frame when viewed from the origin of the Cartesian coordinate system in the positive direction of that axis, that is a positive rotation about the Z-axis only from system 'A' to system 'B' will result in a smaller longitude value for the point in system 'B'.\
\
M : The scale factor to be applied to the position vector in coordinate system 'A' in order to obtain the correct scale of coordinate system 'B'. M = (1+S*10 6), whereby S is the scale correction expressed in parts per million.\
\
In the absence of rotations the two formulas are identical; the difference is solely in the rotations. The name of the second method reflects this.\
\
Note that the same rotation that is defined as positive in the first method is consequently negative in the second and vice versa. It is therefore crucial that the convention underlying the definition of the rotation parameters is clearly understood and is communicated when exchanging datum transformation parameters, so that the parameters may be associated with the correct coordinate transformation method (algorithm).","The same example as for the Position Vector Transformation can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:\
9612,4th-order Polynomial function,,,Ordinate 1 of source evaluation point,Ordinate 2 of source evaluation point,Ordinate 1 of target evaluation point,Ordinate 2 of target evaluation point,A1,A2 * m,A3 * n,A4 * mm,A5 * mn,A6 * nn,A7 * mmm,A8 * mmn,A9 * mnn,A10 * nnn,A11 * mmmm,A12 * mmmn,A13 * mmnn,A14 * mnnn,A15 * nnnn,B1,B2 * m,B3 * n,B4 * mm,B5 * mn,B6 * nn,B7 * mmm,B8 * mmn,B9 * mnn,B10 * nnn,B11 * mmmm,B12 * mmmn,B13 * mmnn,B14 * mnnn,B15 * nnnn,,,,"For TRF_POLYNOMIAL 1000, m=Latitude (degrees) of Source Evaluation Point - 55, and n=longitude of Source Evaluation Point (degrees east of Greenwich)",,1996-09-18 00:00:00,,EPSG,,
9613,NADCON,,Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Used specifically for some NAD27<->NAD83 transformations in USA.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Latitude difference gridded binary file,Longitude difference gridded binary file,,,,1996-09-18 00:00:00,US Coast and geodetic Survey - http://www.ngs.noaa.gov,EPSG,Input expects longitudes to be positive west.,
9614,NTv1,,Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Used specifically for some NAD27<->NAD83 transformations in Canada.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Latitude and longitude difference file,,,,,1997-11-13 00:00:00,Geomatics Canada - Geodetic Survey Division.,EPSG,Superceded in 1997 by NTv2 (transformation method code 9615). Input expects longitudes to be positive west.,
9615,NTv2,,Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Latitude and longitude difference file,,,,,1997-11-13 00:00:00,http://www.geod.nrcan.gc.ca/products/html-public/GSDapps/English/NTv2_Fact_Sheet.html,EPSG,Supercedes NTv1 (transformation method code 9614). Input expects longitudes to be positive west.,
9616,Vertical Offset,,This transformation allows calculation of ordinate in the target system by adding the parameter value to the ordinate value of the point in the source system.,Vertical offset,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,V2 = [(V1 * U1) + (O12 * Uoff)] * (m / U2) where V2 = value in second vertical coordinate system; V1 = value in first system; O12 is the value of the the origin of system 2 in system 1; m is unit direction multiplier (m=1 if both systems are height or both are depth; m=-1 if one system is height and the other system is depth; the value of m is implied through the vertical coordinate system type attribute); U1 U2 and Uoff are unit conversion ratios to metres for systems 1 2 and the offset value respectively.,,1999-11-12 00:00:00,,EPSG,,99.79
9617,Madrid to ED50,,,,,,,A,B,C,D,E,F,G,H,J,,,,,,,,,,,,,,,,,,,,,,,,,"The original geographic coordinate system for the Spanish mainland was based on Madrid 1870 datum, Struve 1860 ellipsoid, with longitudes related to the Madrid meridian. Three polynomial expressions have been empirically derived by El Servicio Geogrßfico del EjΘrcito to convert geographical coordinates based on this system to equivalent values based on the European Datum of 1950 (ED50). The polynomial coefficients derived can be used to convert from Madrid 1870 to ED50. Three pairs of expressions have been derived: each pair is used to calculate the shift in latitude and longitude respectively for (i) a mean for all Spain, (ii) a better fit for the north of Spain, (iii) a better fit for the south of Spain.\
\
The polynomial expressions transformations are:\
\
dLat seconds = A + (B*lon) + (C*lat) + (D*h)\
dLon seconds = (E+F) + (G*lon) + (H*lat) + (J*h)\
\
where latitude lat and longitude lon are in decimal degrees referred to the Madrid 1870 (Madrid) geographic coordinate system and h in metres. E is the longitude (in seconds) of the Madrid meridian measured from the Greenwich meridian; it is the value to be applied to a longitude relative to the Madrid meridian to transform it to a longitude relative to the Greenwich meridan.\
\
The results of these expressions are applied through the formulae:\
Lat(ED50) = Lat(M1870(M)) + dLat\
and Lon(ED50) = Lon(M1870(M)) + dLon.","Input point coordinate system: Madrid 1870 (Madrid) (geographic 3D)\
Latitude = 42 deg 38 min 52.77 sec N = 42.647992 degrees\
Longitude = 3 deg 39 min 34.57 sec E of Madrid \
= +3.659603 degrees from the Madrid meridian.\
Height = 0 m\
\
For the north zone transformation:\
A = 11.3287790\
B = -0.0385200\
C = -0.1674000\
D = 0.0000379\
E = -13276.58\
F = 2.5079425\
G = -0.0086400\
H = 0.835200\
J = -0.0000038\
\
dLat = +4.05 seconds\
\
Then ED50 latitude = 42 deg 38 min 52.77 sec N + 4.05sec\
Then ED50 longitude = 3 deg 39 min 34.57 sec E - 3 deg 41 min 10.54 sec\
= 0 deg 01 min 35.97 sec W of Greenwich.",1999-11-15 00:00:00,Institut de Geomatica; Barcelona,EPSG,,99.284 99.82
9618,Geographical and Height Offsets,,This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.,Latitude offset,Longitude offset,Gravity-related to ellipsoid height,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Lat2 = Lat1 + latitude_offset; Lon2 = Lon1 + longitude_offset; EllipsoidHeight2 = GravityHeight1 + gravity-related_to_ellipsoid_height.,,1999-11-12 00:00:00,,EPSG,,99.79
9619,Geographical Offsets,,This transformation allows calculation of coordinates in the target system by adding the parameter value to the coordinate values of the point in the source system.,Latitude offset,Longitude offset,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,Lat2 = Lat1 + latitude_offset; Lon2 = Lon1 + longitude_offset.,,1999-11-12 00:00:00,,EPSG,,99.79
9620,Norway Offshore Interpolation,,,,,,,,Geod. tfm. code for northern boundary,Geod. tfm. code for southern boundary,,,,,,,,,,,,,,,,,,,,,,,,,,,,Geod. tfm. name for northern boundary,Geod. tfm. name for southern boundary,,,,1999-04-22 00:00:00,"Norwegian Mapping Authority note of 13-Feb-1991 \"Om Transformasjon mellom Geodetiske Datum i Norge\".",EPSG,,
9801,Lambert Conic Conformal (1SP),,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:\
\
E = FE + r sin(theta)\
N = FN + r0 - r cos(theta)\
where\
n = sin lat0\
r = a F t^n k0 for r0, and r\
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m0, lat0, and m2, lat2 where lat1 and lat2 are the latitudes of the standard parallels.\
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t0 and t using lat0 and lat respectively.\
F = m0/(n t1^n)\
theta = n(lon - lon0)\
\
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:\
\
lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}\
lon = theta'/n +lon0\
where\
theta' = arctan[(E - FE)/{r0 -(N - FN)}]\
r' = +/-[(E - FE)^2 + {r0 - (N - FN)}^2]^0.5\
t' = (r'/a k0 F)^(1/n)\
and n, F, and rF are derived as for the forward calculation.","For Projected Coordinate System JAD69 / Jamaica National Grid\
\
Parameters:\
Ellipsoid: Clarke 1866, a = 6378206.400 m., 1/f = 294.97870\
then e = 0.08227185 and e^2 = 0.00676866\
\
Latitude Natural Origin 18 deg 00 min 00 sec N = 0.31415927 rad\
Longitude Natural Origin 77 deg 00 min 00 sec W = -1.34390352 rad\
Scale factor at origin 1.000000\
False Eastings FE 250000.00 m\
False Northings FN 150000.00 m\
\
Forward calculation for: \
Latitude: 17 deg 55 min 55.80 sec N = 0.31297535 rad\
Longitude: 76 deg 56 min 37.26 sec W = -1.34292061 rad\
first gives\
m0 = 0.95136402 t0 = 0.72806411\
F = 3.39591092 n = 0.30901699\
r = 19643955.26 r0 = 19636447.86\
theta = 0.00030374 t = 0.728965259\
\
Then Easting E = 255966.58 m\
Northing N = 142493.51 m\
\
Reverse calculation for the same easting and northing first gives\
\
theta' = 0.000303736\
t' = 0.728965259\
m0 = 0.95136402\
r' = 19643955.26\
\
Then Latitude = 17 deg 55 min 55.800 sec N\
Longitude = 76 deg 56 min 37.260 sec W",1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.1",EPSG,,
9802,Lambert Conic Conformal (2SP),,,Latitude of false origin,Longitude of false origin,Latitude of 1st standard parallel,Latitude of 2nd standard parallel,,Easting at false origin,Northing at false origin,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"To derive the projected Easting and Northing coordinates of a point with geographical coordinates (lat,lon) the formulas for the one standard parallel case are:\
\
E = EF + r sin(theta)\
N = NF + rF - r cos(theta)\
where\
m = cos(lat)/(1 - e^2 sin^2(lat))^0.5 for m1, lat1, and m2, lat2 where lat1 and lat2 are the latitudes of the standard parallels.\
t = tan(pi/4 - lat/2)/[(1 - e sin(lat))/(1 + e sin(lat))]^(e/2) for t1, t2, tF and t using lat1, lat2, latF and lat respectively.\
n = (loge(m1) - loge(m2))/(loge(t1) - loge(t2))\
F = m1/(n t1^n)\
r = a F t^n for rF and r, where rF is the radius of the parallel of latitude of the false origin.\
theta = n(lon - lon0)\
\
The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:\
\
lat = pi/2 - 2arctan{t'[(1 - esin(lat))/(1 + esin(lat))]^(e/2)}\
lon = theta'/n +lon0\
where\
r' = +/-[(E - EF)^2 + {rF - (N - NF)}^2]^0.5 , taking the sign of n\
t' = (r'/aF)^(1/n)\
theta' = arctan [(E- EF)/(rF - (N- NF))]\
and n, F, and rF are derived as for the forward calculation.","For Projected Coordinate System NAD27 / Texas South Central\
\
Parameters:\
Ellipsoid Clarke 1866, a = 6378206.400 metres = 20925832.16 US survey feet\
1/f = 294.97870\
then e = 0.08227185 and e^2 = 0.00676866\
\
First Standard Parallel 28o23'00\"N = 0.49538262 rad\
Second Standard Parallel 30o17'00\"N = 0.52854388 rad\
Latitude False Origin 27o50'00\"N = 0.48578331 rad\
Easting at false origin 2000000.00 US survey feet\
Northing at false origin 0.00 US survey feet\
\
Forward calculation for: \
Latitude 28o30'00.00\"N = 0.49741884 rad\
Longitude 96o00'00.00\"W = -1.67551608 rad\
\
first gives :\
m1 = 0.88046050 m2 = 0.86428642\
t = 0.59686306 tF = 0.60475101\
t1 = 0.59823957 t2 = 0.57602212\
n = 0.48991263 F = 2.31154807\
r = 37565039.86 rF = 37807441.20\
theta = 0.02565177\
\
Then Easting E = 2963503.91 US survey feet\
Northing N = 254759.80 US survey feet\
\
Reverse calculation for same easting and northing first gives:\
theta' = 0.025651765 r' = 37565039.86\
t' = 0.59686306\
\
Then Latitude = 28o30'00.000\"N\
Longitude = 96o00'00.000\"W",1999-04-22 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.1",EPSG,,99.281
9803,Lambert Conic Conformal (2SP Belgium),,,Latitude of false origin,Longitude of false origin,Latitude of 1st standard parallel,Latitude of 2nd standard parallel,,Easting at false origin,Northing at false origin,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"Since 1972 a modified form of the two standard parallel case has been used in Belgium. For the Lambert Conic Conformal (2 SP Belgium), the formulas for the standard two standard parallel case are used except for: \
Easting, E = EF + r sin (theta - alpha)\
Northing, N = NF + rF - r cos (theta - alpha)\
and for the reverse formulas\
lon = ((theta' + alpha)/n) +lon0\
where alpha = 29.2985 seconds.","For Projected Coordinate System Belge l972 / Belge Lambert 72\
\
Parameters:\
Ellipsoid International 1924, a = 6378388 metres\
1/f = 297\
then e = 0.08199189 and e^2 = 0.006722670\
\
First Standard Parallel 49o50'00\"N = 0.86975574 rad\
Second Standard Parallel 51o10'00\"N = 0.89302680 rad\
Latitude False Origin 90o00'00\"N = 1.57079633 rad\
Reverse calculation for same easting and northing first gives:\
theta' = 0.01939192 r' = 548041.03\
t' = 0.35913403\
Then Latitude = 50o40'46.461\"N\
Longitude = 5o48'26.533\"E",1999-04-22 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.1",EPSG,,99.281
9804,Mercator (1SP),,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The formulas to derive projected Easting and Northing coordinates are:\
\
E = FE + a*k0(lon - lon0) \
N = FN + a*k0* ln{tan(pi/4 + lat/2)[(1 - esin(lat))(1 + esin(lat))]^e/2} where symbols are as listed above and logarithms are natural.\
\
The reverse formulas to derive latitude and longitude from E and N values are:\
\
lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi) \
Reverse calculation for same easting and northing first gives :\
t = 1.0534121\
chi = -0.0520110\
\
Then Latitude = 3o00'00.000\"S\
Longitude = 120o00'00.000\"E",1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.2",EPSG,,
9805,Mercator (2SP),,,Latitude of 1st standard parallel,Longitude of natural origin,,,,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The formulas to derive projected Easting and Northing coordinates are:\
\
For the two standard parallel case, k0 is first calculated from\
\
k0 = cos(latSP1)/(1 - e^2*sin^2(latSP1))^0.5 \
\
where latSP1 is the absolute value of the first standard parallel (i.e. positive). \
\
Then, for both one and two standard parallel cases, \
\
E = FE + a*k0(lon - lon0) \
N = FN + a*k0* ln{tan(pi/4 + lat/2)[(1 - esin(lat))(1 + esin(lat))]^e/2} where symbols are as listed above and logarithms are natural.\
\
The reverse formulas to derive latitude and longitude from E and N values are:\
\
lat = chi + (esq/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2chi) \
then natural origin at latitude of 0oN has scale factor k0= 0.74426089\
\
Forward calculation for: \
Latitude 53o00'00.00\"N = 0.9250245 rad\
Longitude 53o00'00.00\"E = 0.9250245 rad\
\
gives Easting E = 165704.29 m \
Northing N = 5171848.07 m\
\
Reverse calculation for same easting and northing first gives :\
t = 0.33639129 chi = 0.92179596\
\
Then Latitude = 53o00'00.000\"N\
Longitude = 53o00'00.000\"E",1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.2",EPSG,,
9806,Cassini-Soldner,,,Latitude of natural origin,Longitude of natural origin,,,,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The formulas to derive projected Easting and Northing coordinates are:\
\
Easting E = FE + nu[A - TA^3/6 -(8 - T + 8C)TA^5/120]\
\
Northing N = FN + M - M0 + nu*tan(lat)*[A^2/2 + (5 - T + 6C)A^4/24]\
\
where A = (lon - lon0)cos(lat)\
T = tan^2(lat)\
C = e2 cos2*/(1 - e2) nu = a /(1 - esq*sin^2(lat))^0.5 \
and M, the distance along the meridian from equator to latitude lat, is given by\
where lat1 is the latitude of the point on the central meridian which has the same Northing as the point whose coordinates are sought, and is found from:\
T = 0.03109120 M = 5496860.24 nu = 31709831.92 M0 = 5739691.12\
\
Then Easting E = 66644.94 links\
Northing N = 82536.22 links\
\
Reverse calculation for same easting and northing first gives :\
e1 = 0.00170207 D = -0.01145875\
T1 = 0.03109544 M1 = 5497227.34\
nu1 = 31709832.34 mu1 = 0.17367306\
phi1 = 0.17454458 rho1 = 31501122.40\
\
\
Then Latitude = 10o00'00.000\"N\
Longitude = 62o00'00.000\"W",1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.3",EPSG,,
9807,Transverse Mercator,,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The formulas to derive the projected Easting and Northing coordinates are in the form of a series as follows:\
\
Easting, E = FE + k0*nu[A + (1 - T + C)A^3/6 + (5 - 18T + T^2 + 72C - 58e'sq)A^5/120] \
For areas south of the equator the value of latitude lat will be negative and the formulas above, to compute the E and N, will automatically result in the correct values. Note that the false northings of the origin, if the equator, will need to be large to avoid negative northings and for the UTM projection is in fact 10,000,000m. Alternatively, as in the case of Argentina's Transverse Mercator (Gauss-Kruger) zones, the origin is at the south pole with a northings of zero. However each zone central meridian takes a false easting of 500000m prefixed by an identifying zone number. This ensures that instead of points in \
different zones having the same eastings, every point in the country, irrespective of its projection zone, will have a unique set of projected system coordinates. Strict application of the above formulas, with south latitudes negative, will result in the derivation of the correct Eastings and Northings. \
\
Similarly, in applying the reverse formulas to determine a latitude south of the equator, a negative sign for lat results from a negative lat1 which in turn results from a negative M1.","For Projected Coordinate System OSGB 1936 / British National Grid\
\
Parameters:\
Ellipsoid Airy 1830 a = 6377563.396 m 1/f = 299.32496\
then e'^2 = 0.00671534 and e^2 = 0.00667054\
\
Latitude Natural Origin 49o00'00\"N = 0.85521133 rad\
Scale factor ko 0.9996013 False Eastings FE 400000.00 m\
False Northings FN -100000.00 m\
\
Forward calculation for: \
Latitude 50o30'00.00\"N = 0.88139127 rad\
Longitude 00o30'00.00\"E = 0.00872665 rad\
A = 0.02775415 C = 0.00271699\
T = 1.47160434 M = 5596050.46\
M0 = 5429228.60 nu = 6390266.03\
\
Then Easting E = 577274.99 m\
Northing N = 69740.50 m\
\
Reverse calculations for same easting and northing first gives :\
e1 = 0.00167322 mu1 = 0.87939562\
M1 = 5599036.80 nu1 = 6390275.88\
phi1 = 0.88185987 D = 0.02775243\
rho1 =6372980.21 C1 = 0.00271391\
T1 = 1.47441726\
\
Then Latitude = 50o30'00.000\"N\
Longitude = 00o30'00.000\"E",1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.4",EPSG,,
9808,Transverse Mercator (South Orientated),,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"For the mapping of southern Africa a south oriented Transverse Mercator projection is used. Here the coordinate axes are called Westings and Southings and increment to the West and South from the origin respectively. The Transverse Mercator formulas need to \
In these formulas the terms FE and FN have been retained for consistency of the terminology. For the reverse formulas, those for the standard Transverse Mercator above apply, with the exception that:\
\
M1 = M0 + (S + FN)/k0\
and D = (W + FE)/(nu1*k0), with nu1 = nu for lat1",,1996-09-18 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.4",EPSG,,
9809,Oblique Stereographic,,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The coordinate transformation from geographical to projected coordinates is executed via the distance and azimuth of the point from the centre point or origin. For a sphere the formulas are relatively simple. For the ellipsoid the parameters defining the conformal sphere at the tangent point as origin are first derived. The conformal latitudes and longitudes are substituted for the geodetic latitudes and longitudes of the spherical formulas for the origin and the point.\
\
Oblique and Equatorial Stereographic Formula\
\
Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are:\
The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from :\
\
chi0 = asin[(w2-1)/(w2+1)]\
\
where S1 and S2 are as above and w2 = c (S1(S2)^e)^n\
\
lambda0 = lon0\
\
For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from \
lambda = n(lon-lambda0) + lambda0\
chi = asin[(w-1)/(w+1)]\
\
where w = c (Ss (Sb)^e)^n\
Sa = (1+sin(lat))/(1-sin(lat))\
Sb = (1-e.sin(lat))/(1+e.sin(lat))\
\
Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)]\
\
N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B\
\
E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B\
\
\
The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.\
\
The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) :\
\
chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)]\
\
lambda = j + 2 i + lambda0\
\
where g = 2 R k0 tan(pi/4 - chi0/2)\
h = 4 R k0 tan(chi0) + g\
i = atan[(E-FE) / {h+(N-FN)}]\
j = atan[(E-FE) / (g-(N-FN)] - i\
\
Geodetic longitude lon = (lambda-lambda0 ) / n + lambda0\
\
Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1- sin(chi))}] / n\
\
First approximation lat1 = 2 atan(e^psi) - pi/2 where e=base of natural logarithms.\
\
psii = isometric latitude at lati\
\
where psii= ln[{tan(lati/2 + pi/4} {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]\
\
Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2)\
\
until the change in lat is sufficiently small.\
\
\
For Oblique Stereographic projections centred on points in the southern hemisphere, the signs of E, N, lon0, lon, must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude.\
\
An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed. The conformal longitude is then always equivalent to the geodetic longitude. This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered to be a different projection method.","For Projected Coordinate System RD / Netherlands New\
\
Parameters:\
Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281\
then e = 0.08169683\
\
Latitude Natural Origin 52o09'22.178\"N = 0.910296727 rad\
where S1 = 8.509582274 S2 = 0.878790173 w1 = 8.428769183\
sin chi0 = 0.787883237\
\
w = 8.492629457 chi0 = 0.909684757 D0 = d0 \
\
for the point chi = 0.924394997 D = 0.104724841\
\
hence B = 1.999870665 N = 557057.739 E = 196105.283\
\
reverse calculation for the same Easting and Northing first gives:\
\
g = 4379954.188 h = 37197327.96 i = 0.001102255 j = 0.008488122\
\
then D = 0.10472467 Longitude = 0.104719584 rad = 6 deg E\
\
chi = 0.924394767 psi = 1.089495123\
phi1 = 0.921804948 psi1 = 1.084170164\
phi2 = 0.925031162 psi2 = 1.089506925\
phi3 = 0.925024504 psi3 = 1.089495505\
phi4 = 0.925024504\
\
Then Latitude = 53o00'00.000\"N\
Longitude = 6o00'00.000\"E",1999-11-15 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.6",EPSG,,99.811
9810,Polar Stereographic,,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"For the forward transformation from latitude and longitude,\
\
E = FE + rho sin(lon - lon0)\
N = FN - rho cos(lon - lon0)\
where\
rho = 2 a ko t /{[((1+e)^(1+e)) ((1-e)^(1-e))]^0.5}\
t = tan (pi/4 - lat/2) / [(1-esin(lat) ) / (1 + e sin(lat))]^(e/2)\
t = rho [((1+e)^(1+e)) ((1-e)^(1-e))]^0.5} / 2 a ko\
rho = [(E-FE)^2 + (N - FN)^2]^0.5",,1996-09-18 00:00:00,"US Geological Survey Professional Paper 1395; \"Map Projections - A Working Manual\"; J. Snyder",EPSG,,
9811,New Zealand Map Grid,,,Latitude of natural origin,Longitude of natural origin,,,,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1996-09-18 00:00:00,New Zealand Department of Lands technical circular 1973/32,EPSG,,
9812,Hotine Oblique Mercator,,,Latitude of projection centre,Longitude of projection centre,Azimuth of initial line,Angle from Rectified to Skew Grid,Scale factor on initial line,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The following constants for the projection may be calculated :\
\
B = (1 + esq * cos^4(latc) / (1 - esq ))^0.5\
A = a * B * kc *(1 - esq )^0.5 / ( 1 - esq * sin^2(latc))\
Latitude Projection Centre fc 4o00'00\"N = 0.069813170 rad\
Longitude Projection Centre lc 115o00'00\"E = 2.007128640 rad\
Azimuth of central line ac 53o18'56.9537\" = 0.930536611 rad\
Rectified to skew gc 53o07'48.3685\" = 0.927295218 rad\
Scale factor ko 0.99984\
False Eastings FE 0.00 m\
False Northings FN 0.00 m\
\
Forward calculation for: \
Latitude f 4o39'20.783\"N = 0.081258569 rad\
Longitude l 114o28'10.539\"E = 1.997871312 rad\
\
B = 1.003303209 F = 1.07212156\
A = 6376278.686 H = 1.00000299\
to = 0.932946976 g0 = 0.92729522\
D = 1.002425787 l0 = 1.91437347\
D2 = 1.004857458\
\
uc = 738096.09 vc = 0.00\
t = 0.922369529 Q = 1.084456854\
S = 0.081168129 T = 1.003288725\
V = 0.83675700 U = 0.014680803\
v = -93307.40 u = 734236.558\
u-uc = -3859.536\
\
Then Easting E = 531404.81 m\
Northing N = 515187.85 m\
\
Reverse calculations for same easting and northing first gives :\
vÆ = -93307.40 uÆ = 734236.558\
uÆ+uc = 1472332.652 QÆ = 1.014790165\
SÆ = 0.014682385 TÆ = 1.000107780 \
VÆ = 0.115274794 UÆ = 0.080902065\
tÆ = 0.922369529 c = 0.080721539 \
\
Then Latitude f = 4o39'20.783\"N \
Longitude l = 114o28'10.539\"E",1999-11-15 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.5",EPSG,,97.62 99.811
9813,Laborde Oblique Mercator,,,Latitude of projection centre,Longitude of projection centre,Azimuth of initial line,,Scale factor on initial line,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1996-09-18 00:00:00,"\"La nouvelle projection du Service Geographique de Madagascar\"; J. Laborde; 1928",EPSG,Can be accomodated by Oblique Mercator method (code 9815).,97.613
9814,Swiss Oblique Cylindrical,,,Latitude of projection centre,Longitude of projection centre,,,,Easting at projection centre,Northing at projection centre,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1996-09-18 00:00:00,"\"Die projecktionen der Schweizerischen Plan und Kartenwerke\"; J Bollinger; 1967",EPSG,Can be accomodated by Oblique Mercator method (code 9815).,97.612
9815,Oblique Mercator,,,Latitude of projection centre,Longitude of projection centre,Azimuth of initial line,Angle from Rectified to Skew Grid,Scale factor on initial line,Easting at projection centre,Northing at projection centre,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"The following constants for the projection may be calculated :\
\
B = (1 + esq * cos^4(latc) / (1 - esq ))^0.5\
A = a * B * kc *(1 - esq )^0.5 / ( 1 - esq * sin^2(latc))\
lon= lon0 - atan ((SÆ cos(gammac) - VÆ sin(gammac)) / cos(B*uÆ / A)) / B",,1999-11-15 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.5",EPSG,,99.811
9816,Tunisia Mining Grid,,,Latitude of origin,Longitude of origin,,,,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"This grid is used as the basis for mineral leasing in Tunsia. Lease areas are approximately 2 x 2 km or 400 hectares. The corners of these blocks are defined through a six figure grid reference where the first three digits are an easting in kilometres and the last three digits are a northing. The latitudes and longitudes for block corners at 2 km intervals are tabulated in a mining decree dated 1st January 1953. From this tabulation in which geographical coordinates are given to 5 decimal places it can be seen that:\
a) the minimum easting is 94 km, on which the longitude is 5.68989 grads east of Paris.\
b) the maximum easting is 490 km, on which the longitude is 10.51515 grads east of Paris.\
c) each 2 km grid easting interval equals 0.02437 grads.\
d) the minimum northing is 40 km, on which the latitude is 33.39 grads.\
e) the maximum northing is 860 km, on which the latitude is 41.6039 grads.\
f) between 40 km N and 360 km N, each 2 km grid northing interval equals 0.02004 grads.\
g) between 360 km N and 860 km N, each 2 km grid northing interval equals 0.02003 grads.\
\
Formulae are:\
\
Grads from Paris\
\
Lat (grads) = 36.5964 + [(N - 360) * A] \
where N is in kilometres and A = 0.010015 if N > 360, else A = 0.01002.\
\
LonParis (grads) = 7.83445 + [(E - 270) * 0.012185], where E is in kilometres.\
\
The reverse formulae are:\
\
E (km) = 270 + [(LonParis - 7.83445) / 0.012185] where LonParis is in grads.\
\
N (km) = 360 + [(Lat - 36.5964) / B] \
where Lat is in grads and B = 0.010015 if lat>36.5964, else B = 0.01002.\
\
Degrees from Greenwich.\
\
Modern practice in Tunisia is to quote latitude and longitude in degrees with longitudes referenced to the Greenwich meridian. The formulae required in addition to the above are:\
\
Lat (degrees) = (Latg * 0.9) where Latg is in grads.\
LonGreenwich (degrees) = [(LonParis + 2.5969213) * 0.9] where LonParis is in grads.\
\
\
Lat (grads) = (Latd / 0.9) where Latd is in decimal degrees.\
LonParis (grads) = [(LonGreenwich / 0.9) - 2.5969213)] where LonGreenwich is in decimal degrees.","For grid location 302598,\
Latitude = 36.5964 + [(598 - 360) * A]. As N > 360, A = 0.010015.\
Latitude = 38.97997 grads = 35.08197 degrees.\
\
Longitude = 7.83445 + [(E - 270) * 0.012185, where E = 302.\
Longitude = 8.22437 grads east of Paris = 9.73916 degrees east of Greenwich.",1999-11-15 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.8",EPSG,,99.811
9817,Lambert Conic Near-Conformal,,,Latitude of natural origin,Longitude of natural origin,,,Scale factor at natural origin,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,"To compute the Lambert Conic Near-Conformal the following formulae are used;\
\
E = FE + r sin(theta)\
N = FN + M + r sin(theta) tan(theta/2) using the natural origin rather than the false origin.\
Then Easting E = 15707.96 m (c.f. E = 15708.00 using full formulae)\
Northing N = 623165.96 m (c.f. N = 623167.20 using full formulae)\
\
Reverse calculation for the same easting and northing first gives\
\
q' = -0.03188875\
rÆ = 8916631.685\
MÆ= 318632.72 \
\
Latitude = 0.654874806 rad = 37d 31' 17.625\" N\
Longitude = 0.595793792 rad = 34d 08' 11.291\" E",1999-11-15 00:00:00,"EPSG Guidance note #7; \"Geographic and Projected Coordinate System Transformations\"; section 1.4.1.1",EPSG,,99.811
9818,American Polyconic,,,Latitude of natural origin,Longitude of natural origin,,,,False easting,False northing,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,1999-10-20 00:00:00,"US Geological Survey Professional Paper 1395; \"Map Projections - A Working Manual\"; J. Snyder",EPSG,,99.55
