EVOLUTION-MANAGER

Edit File: coordinate_operation_method.csv

coord_op_method_code,coord_op_method_name,reverse_op,formula,example,remarks,information_source,data_source,revision_date,change_id,deprecated
1024,Popular Visualisation Pseudo Mercator,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

This method is utilised by some popular web mapping and visualisation applications. It applies standard Mercator (Spherical) formulas (method code 1026) to ellipsoidal coordinates and the sphere radius is taken to be the semi-major axis of the ellipsoid. This approach only approximates to the more rigorous application of ellipsoidal formulas to ellipsoidal coordinates (as given in EPSG dataset coordinate operation method codes 9804 and 9805).  Unlike either the spherical or ellipsoidal Mercator projection methods, this method is not conformal: scale factor varies as a function of azimuth, which creates angular distortion. Despite angular distortion there is no convergence in the meridian.

The formulas to derive projected Easting and Northing coordinates from ellipsoidal latitude (lat) and longitude (lon) first derive the radius of the sphere (R) from:
	R = a

Then applying spherical Mercator formulae:

E = FE + R(lon - lonO)              
	N = FN + R ln[tan(pi/4 + lat/2)]                   
where FE and FN are false easting and false nothing at the projection origin, other symbols are as listed above and logarithms are natural.

If latitude lat = 90º, N is infinite. The above formula for N will fail near to the pole, and should not be used poleward of 88º.

The reverse formulas to derive latitude and longitude on the sphere from E and N values are:
	D = -(N-FN)/R = (FN-N)/R
	lat = pi/2 - 2 atan(e^D) where e=base of natural logarithms, 2.7182818...
	lon = [(E - FE)/R] + lonO

If q_alpha is the scale factor at a given azimuth alpha, it is a function of R', the radius of curvature at that azimuth derived from:
	R' = rho nu / (nu cos^2alpha + rho sin^2alpha)
	q_alpha = R / (R' cos lat)
where rho and nu are the radii of curvature of the ellipsoid at latitude lat in the plane of the meridian and perpendicular to the meridian respectively;
rho = a(1 - e^2)/(1 - e^2 sin^2(lat))^3/2
nu = a /(1 - e^2 sin^2(lat))^1/2

Then when the azimuth is 0º, 180º, 90º or 270º the scale factors in the meridian (h) and on the parallel (k) are:
	q_0 = q_180  = h = R / (rho cos(lat))
	q_90 = q_270 = k = R / (nu cos(lat))
which demonstrates the non-conformallity of the Pseudo Mercator method.

Maximum angular distortion omega is a function of latitude and is found from:
	omega = 2 asin{[ABS(h - k)] / (h + k)}","For Projected Coordinate Reference System: WGS 84 / Pseudo-Mercator

Parameters:
Ellipsoid:	WGS 84	a = 6378137.0 metres		1/f = 298.2572236
			
Latitude of natural origin (latO) = 0°00'00.000""N = 0.0 rad
Longitude of natural origin (lonO) = 0°00'00.000""E = 0.0 rad
False easting (FE) = 0.00 metres	
False northing (FN) = 0.00 metres

Forward calculation for the same coordinate values as used for the Mercator (1SP) (Spherical) example (method code 9841): 
Latitude (lat) = 24°22'54.433""N	= 0.425542460 rad
Longitude (lon) = 100°20'00.000""W = -1.751147016 rad

R  = 6378137.0
whence 
	E = -11 169 055.58 m
	N =   2 800 000.00 m
and 
	h  = 1.1034264
	k  = 1.0972914
	omega = 0°19'10.01""

Reverse calculation for a point 10km north on the grid (-11 169 055.58 m E, 2 810 000.00m N) first gives:
	D = -0.44056752

Then	Latitude (lat) = 0.426970023 rad = 24°27'48.889""N
	Longitude (lon) = -1.751147016 rad = 100°20'00.000""W",Applies spherical formulas to the ellipsoid. As such does not have the properties of a true Mercator projection.,OGP Guidance Note 7-2,OGP,2009/06/07,2008.114 2009.023,0
1025,Geographic3D to GravityRelatedHeight (EGM2008),0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a geoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.

The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.

Calculation of the separation is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the geoid-ellipsoid separation (N) above the ellipsoid of the source Geographic 3D CRS.

Then:
           H = h - N

where h = the height above the ellipsoid in the source geographic 3D CRS
and     H = the geoid height in the target vertical CRS.",(none),"For earlier EGM84 and EGM96 models see Geographic3D to GravityRelatedHeight (EGM), method code 9661.","EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/01/30,2008.097,0
1026,Mercator (Spherical),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The formulas to derive projected Easting and Northing coordinates from spherical latitude lat and longitude lon are:
	E = FE   R (lon - lonO)              
	N = FN   R ln[tan(pi/4   lat/2)]                   
where lonO is the longitude of natural origin and FE and FN are false easting and false nothing.

R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the conformal sphere at the projection origin at latitude latO using the formula for Rc given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the projection properties are not preserved.

If latitude lat = 90º, N is infinite. The above formula for N will fail near to the pole, and should not be used poleward of 88º.

The reverse formulas to derive latitude and longitude on the sphere from E and N values are:
	D = -(N - FN)/R = (FN - N)/R
	lat = pi/2 -  2 atan(e^D) where e=base of natural logarithms, 2.7182818...
	lon = [(E - FE)/R] + lonO

Note that in these formulas, the parameter latitude of natural origin (*O) is not used. However for the Merctor (Spherical) method, for completeness in CRS labelling the EPSG dataset includes this parameter, which must have a value of zero.","For Projected Coordinate Reference System: World Spherical Mercator (Note: CRS not in EPSG dataset)

Parameters:
Sphere:	R = 6371007.0 metres
			
Latitude of natural origin (latO) = 0°00'00.000""N = 0.0 rad
Longitude of natural origin (lonO) = 0°00'00.000""E = 0.0 rad
False easting (FE) = 0.00 metres	
False northing (FN) = 0.00 metres

Forward calculation for: 
Latitude (lat) = 24°22'54.433""N	= 0.425542460 rad
Longitude (lon) = 100°20'00.000""W = -1.751147016 rad

whence 
	E = -11 156 569.90 m
	N =   2 796 869.94 m

Reverse calculation for the same point (-11 156 569.90 m E, 2 796 869.94m N) first gives:
	D = -0.438999665

Then	Latitude (lat) = 0.425542460 rad = 24°22'54.433""N
	Longitude (lon) = -1.751147016 rad = 100°20'00.000""W",,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/06/07,2008.114 2009.023,0
1027,Lambert Azimuthal Equal Area (Spherical),1,"See information source.

R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the authalic sphere using the formula for RA given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the authalic projection properties are not preserved.",See information source.,This is the spherical form of the projection.  See coordinate operation method Lambert Azimuthal Equal Area (code 9820) for ellipsoidal form.  Differences of several tens of metres result from comparison of the two methods.,"USGS Professional Paper 1395, ""Map Projections - A Working Manual"" by John P. Snyder.",OGP,2009/06/07,2009.007 2009.023,0
1028,Equidistant Cylindrical,1,See EPSG Guidance Note #7-2.,See EPSG Guidance Note #7-2.,"See method code 1029 for spherical development. See also Pseudo Plate Carree, method code 9825.",ESRI,OGP,2009/06/07,2009.023,0
1029,Equidistant Cylindrical (Spherical),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

This method has one of the simplest formulas available. If the latitude of natural origin (lat1) is at the equator the method is also known as Plate Carrée. It is not used for rigorous topographic mapping because its distortion characteristics are unsuitable. Formulas are included to distinguish this map projection method from an approach sometimes mistakenly called by the same name and used for simple computer display of geographic coordinates - see Pseudo Plate Carrée (coordinate operation method code 9825).

For the forward calculation:

E =  FE + R . (lon - lonO) . cos(lat1)

N =  FN + R . lat

where lat1, lonO, lat and lon are expressed in radians.

R is the radius of the sphere and will normally be one of the CRS parameters. If the figure of the earth used is an ellipsoid rather than a sphere then R should be calculated as the radius of the conformal sphere at the projection origin at latitude lat1 using the formula for RC given in EPSG Guidance Note 7-2, section 1.2, table 3. Note however that if applying spherical formula to ellipsoidal coordinates, the equidistant projection properties are not preserved.

For the reverse calculation:

lat = (N - FN)/ R

lon = lonO + ([E - FE] / R cos(lat1))

where R is as for the forward method.",See information source.,"See method code 1028 for ellipsoidal development. If the latitude of natural origin is at the equator, also known as Plate Carrée. See also Pseudo Plate Carree, method code 9825.","US Geological Survey Professional Paper 1395; ""Map Projections - A Working Manual"";  J. Snyder.",OGP,2009/06/07,2009.023,0
1030,Geographic3D to GravityRelatedHeight (NZgeoid2009),0,"This transformation involves the application of a geoid-ellipsoid separation value interpolated from a quasigeoid model. The model provides separation values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.

The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.

Then:
           H = h - N

where h = the height above the ellipsoid in the source geographic 3D CRS
and     H = the geoid height in the target vertical CRS.",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/10/17,2009.081,0
1031,Geocentric translations (geocentric domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Xt = Xs + dX;  Yt = Ys + dY;  Zt = Zs + dZ","Given a three parameter datum shift from WGS 84 to ED50 for this North Sea area is given as 
dX = +84.87m, dY = +96.49m, dZ = +116.95m.

The given WGS 84 geocentric values are:

XA = 3771 793.97m
   YA =   140 253.34m
   ZA = 5124 304.35m

Applying the three geocentric translations to these source CRS coordinates we obtain new geocentric values now related to the target CRS ED50:

XB = 3771 878.84m
   YB =   140 349.83m
   ZB = 5124 421.30m",This method allows calculation of geocentric coords in the target system by adding the parameter values to the corresponding coordinates of the point in the source system. See methods 1035 and 9603 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1032,Coordinate Frame Rotation (geocentric domain),1,"<<<<<This text is continued from the description of the Position Vector Transformation formula>>>>>

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Although being common practice particularly in the European E&P industry, the 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:

(Xt)             (  1      +Rz      -Ry)     (Xs)      (dX)
   (Yt)  =  M *  ( -Rz       1      +Rx)  * (Ys)  +  (dY)
   (Zt)              ( +Ry   -Rx        1 )     (Zs)       (dZ)

and the parameters are defined as:

(dX, dY, dZ)   : Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.

(Rx, Ry, Rz)   : Rotations to be applied to the coordinate reference 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 reference frame when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis, that is a positive rotation about the Z-axis only from source coordinate reference system to target coordinate reference system will result in a smaller longitude value for the point in the target coordinate reference system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

M                  : The scale factor to be applied to the position vector in the source coordinate reference system  in order to obtain the correct scale of the target coordinate reference system. M = (1+dS*10^-6), where dS 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 (coordinate operation method 1033) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:

Transformation parameters Coordinate Frame Rotation convention:
dX (m) = 0.000 
dY (m) = 0.000 
dZ (m) = +4.5 
RX ("") = 0.000
RY ("") = 0.000
RZ ("") = -0.554 = -0.000002685868 radians
Scale (ppm) = +0.219

Please note that only the rotation has changed sign as compared to the Position Vector Transformation.",Note the analogy with the Position Vector tfm (code 1033) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1038 and 9607 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1033,Position Vector transformation (geocentric domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The transformation between source and target CRS geocentric coordinates 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:

(Xt)             (  1       -Rz    +Ry)      (Xs)     (dX)
   (Yt)  =  M *  ( +Rz      1      -Rx)  *  (Ys)  + (dY)
   (Zt)              ( -Ry   +Rx       1 )      (Zs)      (dZ)

The parameters are commonly referred to defining the transformation ""from source coordinate reference system to target coordinate reference system"", whereby (Xs, Ys, Zs) are the coordinates of the point in the source geocentric coordinate reference system and (Xt, Yt, Zt) are the coordinates of the point in the target geocentric coordinate reference system.  But 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 with 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 the source coordinate reference system in order to transform from source system to target system; also: the coordinates of the origin of the source coordinate reference system in the target coordinate reference system.

(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 reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system.  Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

M                  :The scale correction to be made to the position vector in the source coordinate reference system in order to obtain the correct scale in the target coordinate reference system. M = (1 + dS*10^-6), where dS is the scale correction expressed in parts per million.

<<<<<This text continues in the description of the Coordinate Frame Rotation formula>>>>>","Input point: 
Coordinate reference system: WGS 72  
Cartesian geocentric coords:
    X = 3 657 660.66 (m)  
    Y =    255 768.55 (m)
    Z = 5 201 382.11 (m)

Transformation parameters WGS 72 to WGS 84:
   dX (m) = 0.000 
   dY (m) = 0.000 
   dZ (m) = +4.5
   RX ("") = 0.000 = 0.0 radians
   RY ("") = 0.000 = 0.0 radians
   RZ ("") = +0.554 = 0.000002685868 radians
   Scale (ppm) = +0.219

Application of this 7 parameter Position Vector Transformation results in WGS 84 geocentric coordinates of:
   X = 3 657 660.78 (m)
   Y =    255 778.43 (m)
   Z =  5 201 387.75 (m)",Note the analogy with the Coordinate Frame Rotation (code 1032) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1037 and 9606 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1034,Molodensky-Badekas (geocentric domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

To eliminate high correlation between the translations and rotations in the derivation of parameter values for the Helmert transformation methods (coordinate operation method codes 1032 and 1033), instead of the rotations being derived about the geocentric coordinate reference system origin they may be derived at a location within the points used in the determination. Three additional parameters, the coordinates of the rotation point, are then required. The formula is:

(Xt)             (  1       +Rz    -Ry)      (Xs - Xp)      (Xp)       (dX)
   (Yt)  =  M *  ( -Rz      1      +Rx)  *  (Ys - Yp)  +  (Yp)  +  (dY)
   (Zt)              ( +Ry   -Rx       1 )      (Zs - Zp)      (Zp)       (dZ)

and the parameters are defined as:

(dX, dY, dZ)   : Translation vector, to be added to the point's position vector in the source coordinate system in order to transform from source coordinate reference system to target coordinate reference system; also: the coordinates of the origin of source coordinate reference system in the target frame.

(Rx, Ry, Rz)   : Rotations to be applied to the coordinate reference 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 reference 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 source coordinate reference system to target coordinate reference system will result in a smaller longitude value for the point in the target coordinate reference system. Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

(Xp, Yp, Zp)   : Coordinates of the point about which the coordinate reference frame is rotated, given in the source Cartesian coordinate reference system.

Reversibility.
The Molodensky-Badekas transformation in a strict mathematical sense is not reversible, i.e. in principle the same parameter values cannot be used to execute the reverse transformation. This is because the evaluation point coordinates are in the forward direction source coordinate reference system and the rotations have been derived about this point. They should not be applied about the point having the same coordinate values in the target coordinate reference system, as is required for the reverse transformation. However, in practical application there are exceptions when applied to the approximation of small differences between the geometry of a set of points in two different coordinate reference systems. The typical vector difference in coordinate values is in the order of 6*10^1 to 6*10^2 metres, whereas the evaluation point on or near the surface of the earth is 6.3*10^6 metres from the origin of the coordinate systems at the Earth's centre. This difference of four or five orders of magnitude allows the transformation in practice to be considered reversible. Note that in the reverse transformation, only the signs of the translations and rotation parameter values are reversed; the coordinates of the evaluation point remain unchanged.","Input point: 
Coordinate reference system: La Canoa

Cartesian geocentric coords:
   XS =  2 550 408.965 m
   YS = -5 749 912.266 m
   ZS =   1 054 891.114 m

Transformation parameters La Canoa to REGVEN:
   dX  = -270.933 m
   dY  = +115.599 m
   dZ  = -360.226 m
   RX   = -5.266 sec = -0.000025530288 radians
   RY  = -1.238 sec = -0.000006001993 radians
   RZ  = +2.381 sec =  0.000011543414 radians
   dS   = -5.109 ppm
   Ordinate 1 of evaluation point  = 2464351.59 m
   Ordinate 2 of evaluation point = -5783466.61 m
   Ordinate 3 of evaluation point =    974809.81 m

Application of the 10 parameter Molodenski-Badekas Transformation results in REGVEN geocentric coordinates of:
   XT =  2 550 138.467 m
   YT = -5 749 799.862 m
   ZT  =  1 054 530.826 m",The Coordinate Frame Rotation is a specific case of the Molodenski-Badekas tfm in which the evaluation point is at the geocentre where geocentric coordinate values are zero. See methods 1039 and 9636 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1035,Geocentric translations  (geog3D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog2D domain) transformation has 3 steps:

(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Geocentric translations(geocentric domain) method, EPSG method code 1031;

(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","(For the geocentric step only). Given a three parameter datum shift from WGS 84 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 WGS 84 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 
to ED50:

XB = 3771 878.84m
   YB =   140 349.83m
   ZB = 5124 421.30m",See methods 1031 and 9603 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1036,Cartesian Grid Offsets from Form Function,1,See information source. The Cartesian offset to be applied is determined through interpolation within an irregular grid of points at which coordinates in both source and target coordinate reference systems are given. The interpolation uses a finite element method form function procedure. The interpolation uses a finite element method form function procedure described in the Information Source.,,Used in German state of Schleswig-Holstein.,"Zeitschrift fur Vermessungswesen (ZfV, the Journal of the German Association of Surveying) volume 128 of April 2003 pages 244-250 and volume 129 of April 2004 pages 258-260.",OGP,2009/11/14,2009.071,0
1037,Position Vector transformation (geog3D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Position Vector(geog3D domain) transformation has 3 steps:

(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Position Vector(geocentric domain) method, EPSG method code 1033;

(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point: 
Coordinate reference system: WGS 72 (geographic 3D)
  Latitude =   55 deg 00 min 00 sec N
  Longitude =  4 deg 00 min 00 sec E
  Ellipsoidal height hS  =  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)

Application of the 7 parameter Position Vector Transformation results in WGS 84 coordinates of:
   X = 3 657 660.78 (m)
   Y =    255 778.43 (m)
   Z = 5 201 387.75 (m)

This converts into:
   Latitude =   55 deg 00 min 00.090 sec N
   Longitude =  4 deg 00 min 00.554 sec E
   Ellipsoidal height =  +3.22 m
on the WGS 84 geographic 3D coordinate reference system.",Note the analogy with the Coordinate Frame Rotation (code 1038) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1033 and 9606 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1038,Coordinate Frame Rotation (geog3D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog3D domain) transformation has 3 steps:

(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Coordinate Frame Rotation (geocentric domain) method, EPSG method code 1032;

(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","The same example as for the Position Vector Transformation (coordinate operation method 1037) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:

Please note that only the rotation has changed sign as compared to the Position Vector Transformation.",Note the analogy with the Position Vector tfm (code 1037) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1032 and 9607 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1039,Molodensky-Badekas (geog3D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Molodensky-Badekas (geog3D domain) transformation has 3 steps:

(i) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(ii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (geocentric domain) method, EPSG method code 1034;

(iii) finally the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602.","Input point: 
Coordinate reference system: La Canoa (geographic 3D)
  Latitude =     9 deg 35 min 00.386 sec N
  Longitude = 66 deg 04 min 48.091 sec W
  Ellipsoidal height hS  =  201.465 m

This transforms to Cartesian geocentric coords:
   XS =  2 550 408.965 m
   YS = -5 749 912.266 m
   ZS =   1 054 891.114 m

Application of the Molodenski-Badekas (geocentric domain) Transformation results in REGVEN geocentric coordinates of:
   XT =  2 550 138.467 m
   YT = -5 749 799.862 m
   ZT  =  1 054 530.826 m

This converts into:
   Latitude =       9 deg 34 min 49.001 sec N
   Longitude =  66 deg 04 min 54.705 sec W
   Ellipsoidal height =  -18.10 m
on the REGVEN geographic 3D coordinate reference system.",See methods 1034 and 9636 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/17,2009.083,0
1040,GNTRANS,1,"GNTRANS is a multistage three-dimensional transformation approach. Starting with a conventional seven-parameter transformation, the next stage functionally models the residuals at identical points using a unique, continuous spline surface. The final stage is a stochastic prediction to describe remaining discrepancies. For application and to improve performance, topocentric coordinate offsets are interpolated from a regular grid derived from the above transformation to determine coordinates of an arbitrary input point in the target system and vice-versa.",,,Geo++.,OGP,2010/03/30,2010.037,0
1041,Krovak (North Orientated),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

From the defining parameters the following constants for the projection may be calculated :

A = a(1 - e^2)^0.5 / [1 - e^2 sin^2.(latC)]
B = {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
gammao = asin[sin(latC) / B]
to = tan(pi/4 + gammao/2).[(1 + e sin(latC)) / (1 - e sin(latC))]^(e.B/2) / [tan(pi/4 + latC/2)]^B
n = sin(latp)
ro = kp.A / tan(latp)

To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:

Then Easting = -Westing
     Northing = -Southing

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
Southing = -Northing
Westing = -Easting
Xp' = Southing  FN
Yp' = Westing  FE
r' = [(Yp')^2 + (Xp')^2]^(1/2)  
theta' = atan[Yp'/Xp']
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))

Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{to^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)

Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK (Ferro) / Krovak East North

Parameters:
Ellipsoid  Bessel 1841   a = 6377397.155m  1/f = 299.15281
    then    e = 0.081696831        e^2 = 0.006674372

Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005

Forward calculation for: 
Latitude = 50°12'32.442""N = 0.876312568 rad
Longitude = 34°30'59.179""E of Ferro = 0.602425500 rad

Then the forward calculation first gives

U = 0.875596951
V = 0.139422687
T = 1.386275051
D = 0.506554626
theta = 0.496385392
r = 1194731.005
Xp = 1050538.634
Yp = 568990.995
Southing = 1050538.634
Westing = 568990.995
 
Then 
Easting  (X) =  -568991.00 m
Northing (Y) = -1050538.64 m

Reverse calculation for the same Easting and Northing:

Southing = 1050538.634
Westing = 568990.995
Xp' = 1050538.634
Yp' = 568990.995
r' = 1194731.005
theta' = 0.496385392
D' = 0.506554626
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603 
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568

Latitude = 0.876312568 rad = 50°12'32.442""N
Longitude = 0.60242500 rad = 34°30'59.179""E of Ferro.",,"Land Survey Office (ZU), Prague. www.cuzk.cz/zu",OGP,2010/12/07,2010.071 2010.100,0
1042,Krovak Modified,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

From the defining parameters the following constants for the projection may be calculated :

To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:

U = 2(atan{to.tan^B(lat/2 + pi/4) / [(1 + e sin(lat)) / (1 - e sin(lat))]^[e.B/2]} - pi/4)
V = B(lonO - lon) where lonO and lon must both be referenced to the same prime meridian.
T = asin[cos(alphaC).sin(U) + sin(alphaC).cos(U). cos(V)]
D = asin[cos(U).sin(V)/cos(T)]
theta = n.D
r = ro.tan^n(pi/4 + latp/2) / tan^n(T/2 + pi/4)
Xp = r.cos(theta)
Yp = r.sin(theta)
Xr = Xp  Xo
Yr = Yp  Yo
dX = C1 + C3.Xr  C4.Yr  2.C6.Xr.Yr + C5.(Xr^2  Yr^2) + C8.Xr.(Xr^2  3.Yr^2)  C7.Yr.(3.Xr^2  Yr^2)  4.C10.Xr.Yr.(Xr^2  Yr^2) + C9.(Xr^4 + Yr^4  6.Xr^2.Yr^2)
dY = C2 + C3.Yr + C4.Xr + 2.C5.Xr.Yr + C6.(Xr^2  Yr^2) + C7.Xr.(Xr^2  3.Yr^2) C8.Yr.(3.Xr^2  Yr^2) + 4.C9.Xr.Yr.(Xr^2  Yr^2) + C10.(Xr^4 + Yr^4  6.Xr^2.Yr^2)
Southing   X = FN + Xp  dX
Westing    Y = FE + Yp  dY

The reverse formulas to derive the latitude and longitude of a point from its Southing and Westing values are:

Xr' = (Southing  FN)  Xo
Yr' = (Westing  FE)  Yo
dX' = C1 + C3.Xr'  C4.Yr'  2.C6.Xr'.Yr' + C5.(Xr'^2  Yr'^2) + C8.Xr'.(Xr'^2  3.Yr'^2)  C7.Yr'.(3.Xr'^2  Yr'^2)  4.C10.Xr'.Yr'.(Xr'^2  Yr'^2) + C9.(Xr'^4 + Yr'^4  6.Xr'^2.Yr'^2)
dY' = C2 + C3.Yr' + C4.Xr' + 2.C5.Xr'.Yr' + C6.(Xr'^2  Yr'^2) + C7.Xr'.(Xr'^2  3.Yr'^2) 
 C8.Yr'.(3.Xr'^2  Yr'^2) + 4.C9.Xr'.Yr'.(Xr'^2  Yr'^2) + C10.(Xr'^4 + Yr'^4  6.Xr'^2.Yr'^2)
Xp' = (Southing  FN) + dX'
Yp' = (Westing  FE) + dY'
r' = [(Yp')^2 + (Xp')^2]^(1/2)  
theta' = atan[Yp'/Xp']
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))

Then latitude lat is found by iteration using U' as the value for lat(j-1) in the first iteration:
lat(j) = 2*(atan{tO^(-1/B) tan^(1/B).(U'/2 + pi/4).[(1 + e sin(lat(j-1)) / (1 - e sin(lat(j-1))]^(e/2)} - pi/4)

Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK/05 (Ferro) / Modified Krovak

Parameters:
Ellipsoid  Bessel 1841   a = 6377397.155m  1/f = 299.15281
    then    e = 0.081696831        e^2 = 0.006674372

Latitude of projection centre = 49°30'00""N =  0.863937979 rad
Longitude of Origin = 42°30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30°17'17.303"" = 0.528627762 rad
Latitude of pseudo standard parallel = 78°30'00""N 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 5000000.00 m
False Northing = 5000000.00 m
Ordinate 1 of evaluation point Xo = 1089000.00 m
Ordinate 2 of evaluation point Yo = 654000.00 m
C1 = 2.946529277E-02
C2 = 2.515965696E-02
C3 = 1.193845912E-07
C4 = -4.668270147E-07
C5 = 9.233980362E-12
C6 = 1.523735715E-12
C7 = 1.696780024E-18
C8 = 4.408314235E-18
C9 = -8.331083518E-24
C10 = -3.689471323E-24

Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005

Forward calculation for: 
Latitude = 50°12'32.442""N = 0.876312568 rad
Longitude = 34°30'59.179""E of Ferro = 0.602425500 rad

U = 0.875596951
V = 0.139422687
S = 1.386275051
D = 0.506554626
theta = 0.496385392
r = 1194731.005
Xp = 1050538.634
Yp = 568990.995
Xr = -38461.366
Yr = -85009.005
dX = -0.075
dY = 0.086
and	
Southing X = 6050538.71 m
Westing	 Y = 5568990.91 m

Reverse calculation for the same Southing and Westing:

Xr' = -38461.291
Yr' = -85009.090
dX' = -0.075
dY' = 0.086
Xp' = 1050538.634
Yp' = 568990.995
r' = 1194731.005
theta' = 0.496385392
D' = 0.506554626
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603 
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568

Latitude = 0.876312568 rad = 50°12'32.442""N
Longitude = 0.294083997 rad = 16°50'59.179""E",Incorporates a polynomial transformation which is defined to be exact and for practical purposes is considered to be a map projection.,"Land Survey Office (ZU), Prague. www.cuzk.cz/zu",OGP,2010/12/07,2010.071 2010.100,0
1043,Krovak Modified (North Orientated),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

From the defining parameters the following constants for the projection may be calculated :

To derive the projected Southing and Westing coordinates of a point with geographical coordinates (lat, lon) the formulas for the Krovak are:

The reverse formulas to derive the latitude and longitude of a point from its Easting and Northing values are:
Southing = -(Northing)
Westing = -(Easting)

Xr' = (Southing - FN)  Xo
Yr' = (Westing - FE)  Yo
dX' = C1 + C3.Xr'  C4.Yr'  2.C6.Xr'.Yr' + C5.(Xr'^2  Yr'^2) + C8.Xr'.(Xr'^2  3.Yr'^2)  C7.Yr'.(3.Xr'^2  Yr'^2)  4.C10.Xr'.Yr'.(Xr'^2  Yr'^2) + C9.(Xr'^4 + Yr'^4  6.Xr'^2.Yr'^2)
dY' = C2 + C3.Yr' + C4.Xr' + 2.C5.Xr'.Yr' + C6.(Xr'^2  Yr'^2) + C7.Xr'.(Xr'^2  3.Yr'^2) 
 C8.Yr'.(3.Xr'^2  Yr'^2) + 4.C9.Xr'.Yr'.(Xr'^2  Yr'^2) + C10.(Xr'^4 + Yr'^4  6.Xr'^2.Yr'^2)
Xp' = (Southing - FN) + dX'
Yp' = (Westing - FE) + dY'
r' = [(Yp')^2 + (Xp')^2]^(1/2)  
theta' = atan[Yp'/Xp']
D' = theta' / sin(latp)
T' = 2{atan[((ro / r')^(1/n)).tan(pi/4 + latp/2)] - pi/4}
U' = asin[cos(alphaC).sin(T') - sin(alphaC).cos(T').cos(D')]
V' = asin(cos(T').sin(D') / cos(U'))

Then
lon = lonO - V' / B where lon is referenced to the same prime meridian as lonO.","For Projected Coordinate Reference System: S-JTSK (Ferro) / Modified Krovak

Parameters:
Ellipsoid  Bessel 1841   a = 6377397.155m  1/f = 299.15281
    then    e = 0.081696831        e^2 = 0.006674372

Latitude of projection centre = 49°30'00""N =  0.863937979 rad
Longitude of Origin = 42°30'00""E of Ferro = 0.741764932 rad
Co-latitude of cone axis = 30°17'17.303"" = 0.528627762 rad
Latitude of pseudo standard parallel = 78°30'00""N = 1.370083463 rad
Scale factor on pseudo Standard Parallel (ko) = 0.9999
False Easting = 0.00 m
False Northing = 0.00 m
Ordinate 1 of evaluation point Xo = 1089000.00 m
Ordinate 2 of evaluation point Yo = 654000.00 m
C1 = 2.946529277E-02
C2 = 2.515965696E-02
C3 = 1.193845912E-07
C4 = -4.668270147E-07
C5 = 9.233980362E-12
C6 = 1.523735715E-12
C7 = 1.696780024E-18
C8 = 4.408314235E-18
C9 = -8.331083518E-24
C10 = -3.689471323E-24

Calculated projection constants:
A = 6380703.611
B = 1.000597498
gammao = 0.863239103
to = 1.003419164
n = 0.979924705
ro = 1298039.005

Forward calculation for: 
Latitude = 50°12'32.442""N = 0.876312568 rad
Longitude = 34°30'59.1790""E of Ferro = 0.602425500 rad

Then the forward calculation first gives

U = 0.875596951
V = 0.139422687
T = 1.386275051
D = 0.506554626
theta = 0.496385392
r = 1194731.005
Xp = 1050538.634
Yp = 568990.995
Xr = -38461.366
Yr = -85009.005
dX = -0.075
dY = 0.086
Southing = 6050538.71 m
Westing	= 5568990.91 m
and then
Easting	X = -5568990.91 m
Northing Y = -6050538.71 m

Reverse calculation for the same Easting and Northing:
Southing = 6050538.71 m
Westing	= 5568990.91 m
Xr' = -38461.290
Yr' = -85009.090
dX' = -0.075
dY' = 0.086
Xp' = 1050538.634
Yp' = 568990.995
r' = 1194731.005
theta' = 0.496385392
D' = 0.506554626
T' = 1.386275051
U' = 0.875596951
V' = 0.139422687
lat(iteration 1) = 0.876310603 
lat(iteration 2) = 0.876312562
lat(iteration 3) = 0.876312568

Latitude = 0.876312568 rad = 50°12'32.442""N
Longitude = 0.602425500 rad = 34°30'59.179""E of Ferro.",Incorporates a polynomial transformation which is defined to be exact and for practical purposes is considered to be a map projection.,"Land Survey Office (ZU), Prague. www.cuzk.cz/zu",OGP,2010/12/07,2010.071 2010.100,0
1044,Mercator (variant C),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The formulas to derive projected Easting and Northing coordinates are:

ko = cos(latSP1)/(1 - e^2*sin^2(latSP1))^0.5 
 
where latSP1 is the absolute value of the first standard parallel (i.e. positive).

E = FE + a*ko(lon - lonF)             
M = a ko ln{tan(pi/4 + latF/2)[(1  e sin(latF))/(1 + e sin(latF))]^(e/2)}
N = (NF  M) + a ko ln{tan(pi/4 + lat/2)[(1  e sin(lat))/(1 + e sin(lat))]^(e/2)}
where 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) 
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4chi)
+ (7e^6/120 +  81e^8/1120) sin(6chi)  + (4279e^8/161280) sin(8chi)

where chi = pi/2 - 2 arctan t
t = B^((NF-M-N)/a*ko) 
B = base of the natural logarithm, 2.7182818...
and ko is calculated as for the forward transformation above.
lon  =  ((E - EF)/a*ko) + lonF","Parameters:
Ellipsoid  Krassowsky 1940   a = 6378245.00m   1/f = 298.300
then e = 0.08181333 and e^2 = 0.00669342

Latitude first SP = 42°00'00""N = 0.73303829 rad
Longitude of natural origin = 51°00'00""E = 0.89011792 rad
Latitude of false origin = 42°00'00""N = 0.73303829 rad
Eastings at false origin EF = 0.00 m
Northing at false origin NF = 0.00 m
  
Forward calculation for: 
Latitude = 53°00'00.00""N = 0.9250245 rad
Longitude = 53°00'00.00""E = 0.9250245 rad

ko = 0.744260894
M = 3819897.85
Easting E = 165704.29 m
Northing N = 1351950.22 m

Reverse calculation for same easting and northing first gives :
t = 0.336391288
chi = 0.921795958
Latitude = 53°00'00.000""N
Longitude = 53°00'00.000""E",,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2010/12/08,2010.058 2010.103,0
1045,Geographic3D to GravityRelatedHeight (OSGM02-Ire),0,"This transformation involves the application of a height difference interpolated from a height correction model. The model provides height difference values at the nodes on a regular grid of latitude and longitude intersection points. The geodetic latitude and longitude used to interpolate within the grid are not affected by this transformation.

The grid is referenced to a specific geographic CRS (the source CRS) and interpolation must be made in this system.

Calculation of the height difference is achieved through a bi-linear interpolation of the grid, using the latitude and longitude of the point. This step provides the height correction (C) of the target datum above the ellipsoid of the source Geographic 3D CRS. C differs from the geoid-ellipsoid separation N because a vertical datum is a realisation of the geoid surface, not the geoid itself.

Then:
           H = h - C

where h = the height above the ellipsoid in the source geographic 3D CRS
and     H = the gravity-related height in the target vertical CRS.",(none),Transformation of the vertical component of a Geographic 3D CRS to a Vertical CRS.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2011/01/14,2010.104,0
9601,Longitude rotation,1,Target_longitude = Source_longitude + longitude_offset.,(none),This transformation allows calculation of the longitude of a point in the target system by adding the parameter value to the longitude value of the point in the source system.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,1999/11/12,1999.790,0
9602,Geographic/geocentric conversions,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Latitude, P, and Longitude, L, in terms of Geographic Coordinate Reference System A may 
be expressed in terms of a geocentric (earth centred) Cartesian coordinate reference 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 prime meridian for geogCRS A is not 
Greewich, longitudes must first be transformed to their Greenwich equivalent. If the earth's 
spheroidal semi major axis is a, semi minor axis  b, and inverse flattening 1/f,  then

X =   (nu + h) cos P cos L
   Y =   (nu + h) cos P sin L
   Z =  ((1 - e^2) nu + h) 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 reference system B may be used to derive geographical coordinates in terms of geographic coordinate reference system B by:
           
P = atan[(Z + eta b sin^3 q) / (p  e^2 a cos^3 q)]
L  = atan (Y/X)

where
eta = e^2 / (1  e^2)
b = a(1  f)
p = (X^2 + Y^2)^0.5
q = atan[(Z a) / (p b)]

and where L 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.

Then
  	h (p / cos P)  nu

(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 +80m in the North Atlantic.)","Consider a North Sea point with coordinates derived by GPS satellite in the WGS 84 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 WGS 84 to ED50 for this North Sea area is given as dX = +84.87m, dY = +96.49m, dZ = +116.95m.

The WGS 84 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.","This is a parameter-less conversion. In applications it is often concatenated with the 3- 7- or 10-parameter transformations 9603, 9606, 9607 or 9636 to form a geographic to geographic transformation.","EPSG guidance note #7-2, http://www.epsg.org, from ""Datums and Map Projections""; Iliffe and Lott (2007).",OGP,2007/11/01,1997.290 2002.510 2004.330 2007.058,0
9603,Geocentric translations (geog2D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog2D domain) transformation has 5 steps:

(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;

(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Geocentric translations(geocentric domain) method, EPSG method code 1031;

(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
 
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","(For the geocentric step only). Given a three parameter datum shift from WGS 84 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 WGS 84 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 
to ED50:

XB = 3771 878.84m
   YB =   140 349.83m
   ZB = 5124 421.30m",See methods 1031 and 1035 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/24,2009.083,0
9604,Molodensky,1,See information source.,(none),See Abridged Molodensky.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2004/08/25,2004.470,0
9605,Abridged Molodensky,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

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 geographic 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:

dlat "" = [(-dX*sin(lat)*cos(lon)) - (dY*sin(lat)*sin(lon)) + (dZ*cos(lat)) + (((a*Df) + (f*Da))*sin(2*lat))] / (rho * sin(1""))

dlon "" = (-dX*sin(lon) + dY*cos(lon)) / ((nu*cos(lat)) * sin(1""))

dh = (dX*cos(lat)*cos(lon)) + (dY*cos(lat)*sin(lon)) + (dZ*sin(lat)) + ((a*Df + f*Da)*(sin(lat)^2)) - da

where the dX, dY and dZ terms are the geocentric translation parameters, and rho and nu are the meridian and prime vertical radii of curvature at the given latitude (lat) on the first ellipsoid, 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.","For a North Sea point with coordinates derived by GPS satellite in the WGS84 geographical coordinate reference system, with coordinates of:
           latitude lat_s                =53°48'33.82""N, 
           longitude lon_s             = 2°07'46.38""E, 
    and ellipsoidal height h_s = 73.0m,

whose coordinates are required in terms of the ED50 geographical coordinate reference system which takes the International 1924 ellipsoid.

The three geocentric translations parameter values from WGS84 to ED50 for this North Sea area are given as dX = +84.87m, dY = +96.49m, dZ = +116.95m. 
Ellipsoid Parameters are:
WGS 84               a = 6378137.0 metres    1/f = 298.2572236
International 1924  a = 6378388.0 metres    1/f = 297.0

Then
da = 6378388  6378137 = 251
df = 0.003367003 - 0.003352811  = 1.41927E-05
whence
dlat  = 2.543""
dlon  = 5.097""
dh   =   44.909 m

ED50 values on the International 1924 ellipsoid are then:
           latitude lat_t                 = 53°48'36.563""N, 
           longitude  lon_t             =  2°07'51.477""E, 
    and ellipsoidal height h_t  = 28.091 m.","This transformation is a truncated Taylor series expansion of a transformation between two geographic coordinate systems, modelled as a set of geocentric translations.","EPSG guidance note #7-2, http://www.epsg.org",OGP,2006/06/12,1999.010 2004.470 2006.420,0
9606,Position Vector transformation (geog2D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog2D domain) transformation has 5 steps:

(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;

(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Position Vector(geocentric domain) method, EPSG method code 1033;

(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
 
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point: 
Coordinate reference system: WGS 72 (geographic 2D)
  Latitude =   55 deg 00 min 00 sec N
  Longitude =  4 deg 00 min 00 sec E
  
This is taken to be geographic 3D with an assumed Ellipsoidal height hS  =  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)

Application of the 7 parameter Position Vector Transformation results in WGS 84 coordinates of:
   X = 3 657 660.78 (m)
   Y =    255 778.43 (m)
   Z = 5 201 387.75 (m)

This converts into:
   Latitude =   55 deg 00 min 00.090 sec N
   Longitude =  4 deg 00 min 00.554 sec E
   Ellipsoidal height =  +3.22 m
on the WGS 84 geographic 3D coordinate reference system. For the 2D equivalent the height is ignored.",Note the analogy with the Coordinate Frame Rotation (code 9607) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1033 and 1037 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/24,1998.160 2009.083,0
9607,Coordinate Frame Rotation (geog2D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog2D domain) transformation has 5 steps:

(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;

(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Coordinate Frame Rotation (geocentric domain) method, EPSG method code 1032;

(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
 
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","The same example as for the Position Vector Transformation (coordinate operation method 1033) can be calculated, however the following transformation parameters have to be applied to achieve the same input and output in terms of coordinate values:

Please note that only the rotation has changed sign as compared to the Position Vector Transformation.",Note the analogy with the Position Vector tfm (code 9606) but beware of the differences!  The Position Vector convention is used by IAG and recommended by ISO 19111. See methods 1032 and 1038 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/24,2004.141 2009.083,0
9613,NADCON,1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. Separate grid files are given for latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.

Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.",(none),Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation.  Input expects longitudes to be positive west.,US Coast and geodetic Survey - http://www.ngs.noaa.gov,OGP,2008/04/11,2004.190 2008.027,0
9614,NTv1,1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. One grid file contains both latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.

Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.",(none),Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation. Superseded in 1997 by NTv2 (transformation method code 9615).  Input expects longitudes to be positive west.,Geomatics Canada - Geodetic Survey Division.,OGP,2008/04/11,2004.190 2008.027,0
9615,NTv2,1,"The latitude and longitude offsets at a point are derived by interpolation within the gridded data. One grid file contains both latitude and longitude offsets. The grid file format is given in documentation available from the information source. Bi-linear interpolation is used to derive the offset values. For the forward calculation the interpolated value of the offset is then added to the source CRS coordinate value to give the coordinates in the target CRS.

Reversibility
Iteration is required for the reverse transformation. The coordinate reference system for the coordinates of the grid nodes is the source coordinate reference system for the forward transformation. Then in forward transformations the offset is obtained through straightforward interpolation of the grid file. But for the reverse transformation the first grid interpolation entry will be the value of the point in the second coordinate reference system, the offsets are interpolated and applied with sign reversed, and the result used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant.",(none),Geodetic transformation operating on geographic coordinate differences by bi-linear interpolation.  Supersedes  NTv1 (transformation method code 9614).  Input expects longitudes to be positive west.,"Geomatics Canada - Geodetic Survey Division, http://www.geod.nrcan.gc.ca/.  Also GDAit software documentation from University of Melbourne Geomatics Department, http://www.sli.unimelb.edu.au/gda94/.",OGP,2008/04/11,2004.190 2008.027,0
9616,Vertical Offset,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

A vertical offset allows calculation of coordinates in the target vertical coordinate reference system by adding a correction parameter A to the coordinate values of the point in the source system:
X2 = X1 + A1>2
where
X2 = value in the forward target vertical coordinate reference system.
X1 = value in the forward source vertical coordinate reference system.
A1>2 is the offset to be applied for the transformation from CRS 1 to CRS 2. Its value for the forward calculation is the value of the origin of the source CRS 1 in the target CRS 2.

For the reverse transformation from CRS 2 to CRS 1 the same formula is used but with the sign of the offset A1>2 reversed:
	X1 = X2 + (A1>2)

Change of axis direction
The above formulas apply only when the positive direction of the axis of each CRS is the same. If there is a requirement to transform heights in the source CRS to depths in the target CRS or to transform depths in the source CRS to heights in the target CRS, the formulas must be modified to:
for the forward transformation:       X2 = mX1 + A1>2
for the reverse transformation:        X1 = m[X2 + (A1>2)]
where m is a direction modifier,
m = p1 * p2 
and
	p1 indicates the positive direction of the CRS 1 axis; p1 = +1 if up, -1 if down;.
	p2 indicates the positive direction of the CRS 2 axis; p2 = +1 if up, -1 if down
These modified formulas remain valid whether or not there is a change in axis direction.

Change of unit
A further modification allows for source CRS axis, target CRS axis or offset to be in different units giving the general formulas:
for the forward transformation:       X2 = {(p1 * p2) * (X1 * U1) + (A1>2 * UA)} / U2
for the reverse transformation:        X1 = {(p1 * p2) * [(X2 * U2) + (A1>2 * UA)]} / U1

where U1 U2 and UA are unit conversion ratios for the two systems and the offset value respectively. U = [(factor b) / (factor c)] from the EPSG Dataset Unit of Measure table, populated with respect to the linear base unit, metre. U has a value of 0.3048 for the international foot.",(none),This transformation allows calculation of height (or depth) in the target system by adding the parameter value to the height (or depth)-value of the point in the source system.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2008/04/11,1999.790 2008.010,0
9617,Madrid to ED50 polynomial,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The polynomial expressions are:

dLat seconds = A0 + (A1*lat) + (A2*lon) + (A3*H)
      dLon seconds = B00 + B0 + (B1*lat) + (B2*lon) + (B3*H)

where latitude lat and longitude lon are in decimal degrees referred to the Madrid 1870 (Madrid) geographic coordinate reference system and H is gravity-related height in metres.  B00 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:
A0 = 11.328779
A1 = -0.1674
A2 = -0.03852
A3 = 0.0000379
B00 = -13276.58
B0 = 2.5079425
B1 = 0.08352
B2 = -0.00864
B3 = -0.0000038

dLat = +4.05 seconds

Then ED50 latitude = 42 deg 38 min 52.77 sec N + 4.05sec = 42 deg 38 min 56.82 sec N

dLon = -13270.54 seconds  = -3 deg 41 min 10.54 sec

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.",,"EPSG guidance note #7-2, http://www.epsg.org, after Institut de Geomatica; Barcelona.",OGP,2011/07/10,1999.284 1999.820 1999.640 2006.910 2011.003 2011.057,0
9618,Geographic2D with Height Offsets,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Lat_T = Lat_S + latitude_offset 
Lon_T = Lon_S + longitude_offset 
EllipsoidHeight_T = GravityHeight_S + gravity-related_to_ellipsoid_height_offset.",(none),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.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2004/04/27,1999.790 2004.330,0
9619,Geographic2D offsets,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Lat_T = Lat_S + latitude_offset 
Lon_T = Lon_S + longitude_offset.","A position with coordinates of 38°08'36.565""N, 23°48'16.235""E referenced to the old Greek geographic 2D coordinate reference system (EPSG CRS code 4120) is to be transformed to the newer GGRS87 system (EPSG CRS code 4121). Transformation parameters from Greek to GGRS87 are:
dLat	=	-5.86""
dLon	=	+0.28""

Then Lat(GGRS87) 	 =	38°08'3656 5"" N		(5.8 6"")	 =	38°08'30.705""N
and Lon(GGRS87) 	=	23°48'16.23  5""E	+	0. 28""	 =	23°48'16.515""E

For the reverse transformation for the same point, 
Lat(GREEK) 	 =	38°08'30.705  "" N	+	5. 86""  	 =	38°08'36.565""N
Lon(GREEK) 	 =	23°48'16.515"" E 	 +	(-0. 28 "")	 =	23°48'16.235""E",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.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2004/04/27,1999.790 2004.330,0
9620,Norway Offshore Interpolation,0,See information source.,(none),"Although in principle this method is not reversible, in practice reversibility is better than 10 cm. For the applications for which it was designed it may be considered reversible.","Norwegian Mapping Authority note of 13-Feb-1991 ""Om Transformasjon mellom Geodetiske Datum i Norge"".",OGP,2005/05/21,2005.230,0
9621,Similarity transformation,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The similarity transformation in algebraic form is:

XT = XT0  + XS * M * cos q  + YS * M * sin q
YT = YT0   XS * M * sin q  + YS * M * cos q

where:
XT0 , YT0    =   the coordinates of the origin point of the source coordinate reference system expressed in the target coordinate reference system;
M                 =  the length of one unit in the source coordinate reference system expressed in units of the target coordinate reference system;
q                  = the angle about which the axes of the source coordinate reference system need to be rotated to coincide with the axes of the target coordinate reference system, counter-clockwise being positive. Alternatively, the bearing of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.

The similarity transformation can also be described as a special case of the parametric affine transformation where coefficients A1 = B2  and  A2 =  - B1.

Reversibility
The reverse formula for the Similarity Transformation is:

XS = [(XT   XTO) * cos q     (YT  YTO) * sin q ] / [M ]
YS = [(XT    XTO) * sin q   +  (YT  YTO) * cos q] / [M ]

An alternative approach for the reverse operation is to use the same formula as for the forward computation but with different parameter values:
	XT = XTO'  + XS * M' * cos theta'  + YS * M' * sin theta'
	YT = YTO'    XS * M' * sin theta' + YS * M' * cos theta'

The reverse parameter values, indicated by a prime ('), can be calculated from those of the forward operation as follows:
XTO' =  (YTO sin theta   XTO cos theta) / M
YTO' =  (YTO cos theta +  XTO sin theta) / M
M'   =  1/M
theta'    =  theta","ED50 / UTM zone 31N to ETRS89 / UTM zone 31N

Parameters of the Similarity Transformation:
XTO   = -129.549 metres
YTO   = -208.185 metres
M     = 1.00000155
theta = 1.56504"" = 0.000007588 rad

Forward computation for source coordinates 300000m E, 4500000m N:

E(ETRS89) = 129.549 + 300000.465 + 34.144
          = 299905.060 m E

N(ETRS89) = 208.185 2.276 + 4500006.977
          = 4499796.515m N

Reverse computation of ETRS89 / UTM 31N coordinates 299905.060m E, 4499796.515m N:

E(ED50) = (300034.609  34.144) / 1.00000155
	= 300000.000m E

N(ED50) = (2.276 + 4500004.700) / 1.00000155
	= 4500000.000m N

Alternative reverse computation:

First calculate new parameter values:
XTO'   = 129.5472 m
YTO'   = 208.1857 m
M'     = 0.99999845
theta' = 0.000007588 rad

Then apply these values to forward formula:
E(ED50) = 129.547 + 299904.595 + (34.142)
        = 300000.000 m

N(ED50) = 208.186  (2.276) + 4499789.539
        = 4500000.000 m",Defined for two-dimensional coordinate systems.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2010/06/29,2000.830 2004.670 2010.053,0
9622,Affine orthogonal geometric transformation,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

XT = XT0   +   XS .  k . dSX . cos q   +   YS .  k .  dSY  . sin q
YT = YT0      XS .  k .  dSX . sin q    +   YS .  k .  dSY  . cos q

where:

XT0 ,YT0  = the coordinates of the origin point of the source coordinate reference system, expressed in the target coordinate reference system;
dSX , dSY  = the length of one unit of the source  axis, expressed in units of the target axis, for the X axes and the Y- axes respectively;
k = point scale factor of the target coordinate reference system in a chosen reference point;
q  = the angle through which the source coordinate reference system axes must be rotated to coincide with the target coordinate refderence system axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.","Source coordinate system: imaginary 3D seismic acquisition bin grid.  The two axes are orthogonal, but the unit on the I-axis is 25 metres, whilst the unit on the J-axis is 12.5 metres.
The target projected coordinate system is WGS 84 / UTM Zone 31N and the origin of the bin grid (centre of bin 0,0) is defined at E = 456781.0, N = 5836723.0.  The projected coordinate system point scale factor at the bin grid origin is 0.99984.
The map grid bearing of the I and J axes are 110* and 20* respectively.  Thus the angle through which both the positive I and J axes need to be rotated to coincide with the positive Easting axis and Northing axis respectively is +20 degrees.

Hence: 
XT0 ,	=    456 781.0 m
YT0	= 5 836 723.0 m
dSX 	= 25
dSY	= 12.5
k 	= 0.99984
q	= +20 degrees

Forward calculation for centre of bin with coordinates: I = 300, J = 247:

XT = Easting   = XT0   +   XS . k . dSX . cos q   +   YS . k . dSY  . sin q    = 464 855.62 m.

YT = Northing = YT0      XS . k . dSX . sin q    +   YS . k . dSY  . cos q  = 5 837 055.90 m

Reverse calculation for this point:
XS = [( XT   XT0) . cos qY    (YT  YT0) . sin qY ] / [k . dSX  . cos (qX  qY)] = 230 bins

YS = [(XT    XT0) . sin qX   +  (YT  YT0) . cos qX ] / [k . dSY . cos (qX  qY)]  = 162 bins",,"EPSG guidance note #7, http://www.epsg.org",OGP,2000/06/10,,1
9623,Affine geometric transformation,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

General case.

The geometric representation of the affine transformation is:
XT = XT0   +  XS * k * MX * cos qX   +  YS * k * MY * sin qY
YT = YT0    XS * k * MX *  sin qX   +  YS * k * MY * cos qY 
where:

XT0 ,YT0  = the coordinates of the origin point of the source coordinate reference system, expressed in the target coordinate reference system;
MX , MY  = the length of one unit of the source axis, expressed in units of the target axis, for the first and second source and target axis pairs respectively;
qX , qY   = the angles about which the source coordinate reference system axes XS and YS must be rotated to coincide with the target coordinate reference system axes XT and YT respectively (counter-clockwise being positive).
k = point scale factor of the target coordinate reference system in a chosen reference point;
 
Comparing the algebraic representation with the parameters of the parameteric form (code 9624) it can be seen that the parametric and geometric forms of the affine transformation are related as follows:
A0  =  XT0
A1  = k * MX * cos qX  
A2  = k * MY * sin qY
B0  =  YT0
B1  =    k * MX * sin qX
B2  =   k *MY * cos qY

Reversibility
For the Affine Geometric Transformation, the reverse operation can be described by a different formula, as shown below, in which the same parameter values as the forward transformation may be used:

XS = [( XT   XT0) . cos qY    (YT  YT0) . sin qY ] / [k * MX  * cos (qX  qY)]
YS = [(XT    XT0) . sin qX   +  (YT  YT0) . cos qX ] / [k * MY * cos (qX  qY)]

Orthogonal case

If the source coordinate reference system happens to have orthogonal axes, that is both axes are rotated through the same angle to bring them into the direction of the orthogonal target coordinate reference system axes, i.e. qX = qY = q, then the Affine Geometric Transformation can be simplified to:

XT = XT0   +   XS .  k . MX . cos q   +   YS .  k .  MY  . sin q
YT = YT0      XS .  k .  MX . sin q    +   YS .  k .  MY  . cos q

where:
q  = the angle through which the source coordinate reference system axes must be rotated to coincide with the target coordinate refderence system axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.

The reverse formulas of the general case can also be simplified by replacing qX and qY with q:

XS = [(XT   XTO) * cos q     (YT  YTO) * sin q ] / [k * MX ]
YS = [(XT    XTO) * sin q   +  (YT  YTO) * cos q] / [k * MY ]

In the EPSG dataset this orthogonal case (code 9622) has been deprecated. The formulas for the general case should be used, inserting q for both qX  and qY. The case has been documented here as part of the progression through increasing constraints on the degrees of freedom between the general case and the Similarity Transformation.",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2005/08/26,2004.670,0
9624,Affine parametric transformation,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

XT   =  A0  +  A1 * XS  +  A2 * YS
YT   =  B0  +  B1 * XS  +  B2 * YS
where
XT , YT  are the coordinates of a point P in the target coordinate reference system;
XS , YS   are the coordinates of P in the source coordinate reference system.

Reversibility
The reverse transformation is another affine transformation using the same formulas but with different parameter values.  The reverse parameter values, indicated by a prime (), can be calculated from those of the forward transformation as follows:

D    = A1 * B2      A2 * B1
A0 = (A2 * B0      B2 * A0) / D
B0 = (B1 * A0      A1 * B0) / D
A1 = +B2 / D
A2 =  A2 / D
B1 =  B1 / D
B2 = +A1 / D

Then
	XS   =  A0'  +  A1' * XT  +  A2' * YT
		YS   =  B0'  +  B1' * XT  +  B2' *",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2005/08/23,2004.670,0
9625,General polynomial (2nd-order),0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to manageable numbers, between 10 and +10 at most.

U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0

Then (XT - XT0) = (XS - XS0) + dX
         (YT - YT0) = (YS - YS0) + dY 
or
XT  = XS - XS0 + XT0 + dX
YT  = YS - YS0 + YT0 + dY

where
XT , YT   are coordinates in the target coordinate reference system,
XS , YS  are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0  are coordinates of the evaluation point in the target coordinate reference system.

and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2
dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2",,,EPSG guidance note #7.,OGP,2000/03/07,,1
9626,General polynomial (3rd-order),0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to manageable numbers, between 10 and +10 at most.

U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0

Then (XT - XT0) = (XS - XS0) + dX
         (YT - YT0) = (YS - YS0) + dY 
or
XT  = XS - XS0 + XT0 + dX
YT  = YS - YS0 + YT0 + dY

and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2 + A6.U3 + A7.U2.V + A8.U.V2 + A9.V3
dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2 + B6.U3 +B7.U2.V +B8.U.V2 +B9.V3",,,EPSG guidance note #7.,OGP,2000/03/07,,1
9627,General polynomial (4th-order),0,"The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the input parameters, usually coordinate offsets U and V relative to a central evaluation point, to manageable numbers, between 10 and +10 at most.

U = XS - XS0 in defined units (which may not be those of the coordinate reference system),
V = YS - YS0

Then (XT - XT0) = (XS - XS0) + dX
         (YT - YT0) = (YS - YS0) + dY 
or
XT  = XS - XS0 + XT0 + dX
YT  = YS - YS0 + YT0 + dY

and where
dX = A0 + A1.U + A2.V + A3.U2 + A4.U.V + A5.V2 + A6.U3 + A7.U2.V + A8.U.V2 + A9.V3 + A10.U4 + A11.U3.V + A12.U2.V2 + A13.U.V3 + A14.V4

dY = B0 + B1.U + B2.V +B3.U2 +B4.U.V +B5.V2 + B6.U3 +B7.U2.V +B8.U.V2 +B9.V3 + B10.U4 + B11.U3.V + B12.U2.V2 + B13.U.V3 + B14.V4",,,EPSG guidance note #7.,OGP,2000/03/07,,1
9628,Reversible polynomial (2nd-order),1,See EPSG Guidance Note 7.,,Reversibility is subject to constraints.  See Guidance Note 7 for clarification.,EPSG guidance note #7.,OGP,2000/03/07,1999.640,1
9629,Reversible polynomial (3rd-order),1,See EPSG Guidance Note 7.,,Reversibility is subject to constraints.  See Guidance Note 7 for clarification.,EPSG guidance note #7.,OGP,2000/03/07,1999.640,1
9630,Reversible polynomial (4th-order),1,See EPSG Guidance Note 7.,"For geodetic transformation ED50 to ED87 (1)

Offset unit:  degree
Ordinate 1 of evaluation point X0  =   55° 00' 00.000""N  = +55 degrees
Ordinate 2 of evaluation point Y0  =     0° 00' 00.000""E   =   +0 degrees

Parameters:
A0 = -5.56098E-06   A1 = -1.55391E-06   ...   A14 = -4.01383E-09
B0 = +1.48944E-05   B2 = +2.68191E-05  ...   B14 = +7.62236E-09

Forward calculation for: 
ED50 Latitude     = Xs =52* 3030""N   =     +52.508333333 degrees
ED50 Longitude  = Ys =  2*E=      +2.0 degrees

U = XS - X0 =  * ED50 - X0  = 52.508333333 - 55.0 = -2.491666667 degrees
V = YS - Y0 =  * ED50 - Y0   = 2.0 - 0.0 = 2.0 degrees

dX = A0 + A1.U + ... + A14.V4
      = -5.56098E-06 + (-1.55391E-06 * -2.491666667) + ... + (-4.01383E-09 * 2.0^4)
      = -3.12958E-06 degrees

dY = B0 + B1.U + ... + B14.V4
      = +1.48944E-05 + (2.68191E-05 * -2.491666667) + ... + (7.62236E-09 * 2.0^4)
      = +9.80126E-06 degrees

Then  ED87 Latitude  =   XT = XS + dX
                                  =  52.508333333 - 3.12958E-06   degrees
                                  = 52* 30 29.9887"" N

ED87 Longitude  =   YT = YS + dY
                           = 2* 00 00.0353"" E

Reverse calculation for transformation ED50 to ED87 (1).
The transformation method for the ED50 to ED87 (1) transformation, 4th-order reversible polynomial, is reversible. The same formulas may be applied for the reverse calculation, but coefficients A0 through A14 and B0 through B14 are applied with reversal of their signs. Sign reversal is not applied to the coordinates of the evaluation point. Thus:
Ordinate 1 of evaluation point X0  =   55° 00' 00.000""N  = +55 degrees
Ordinate 2 of evaluation point Y0  =     0° 00' 00.000""E   =   +0 degrees
A0  = +5.56098E-06   A1 = +1.55391E-06   ...   A14 = +4.01383E-09
B0  = -1.48944E-05    B1 = -2.68191E-05    ...   B14 = -7.62236E-09

Reverse calculation for: 
ED87 Latitude     = XS = 52° 3029.9887""N   =     +52.5083301944 degrees
ED87 Longitude  = YS =   2° 00 00.0353"" E   =     +2.0000098055 degrees

U = 52.5083301944 - 55.0 = -2.4916698056 degrees
V = 2.0000098055 - 0.0 = 2.0000098055 degrees

dX = A0 + A1.U + ... + A14.V4
      = +5.56098E-06 + (1.55391E-06 * -2.491666667) + ... + (4.01383E-09 * 2.0000098055^4)
      = +3.12957E-06 degrees

dY = B0 + B1.U + ... + B14.V4
      = -1.48944E-05 + (-2.68191E-05 * -2.491666667) + ... + (-7.62236E-09 * 2.0000098055^4)
      = -9.80124E-06 degrees

Then ED50 Latitude  =   XT = XS + dX
                                 = 52.5083301944 + 3.12957E-06   degrees
                                 = 52° 30 30.000"" N

ED50 Longitude  =   YT = YS + dY
                           = 2° 00 00.000"" E",Reversibility is subject to constraints.  See Guidance Note 7 for clarification.,EPSG guidance note #7.,OGP,2000/03/07,1999.640,1
9631,Complex polynomial (3rd-order),0,"The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the A and B coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A third-order polynomial in complex numbers is used in Belgium.  A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.

(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3

where U = (XS - XS0).10-5
and     V = (YS - YS0).10-5

Then
XT  = XS - XS0 + XT0 + dX
YT  = YS - YS0 + YT0 + dY

where
XT , YT      are coordinates in the target coordinate reference system,
XS , YS      are coordinates in the source coordinate reference system,
XS0 , YS0   are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0   are coordinates of the evaluation point in the target coordinate reference system.

Note that the zero order coefficients of the general polynomial, A0 and B0, have apparently disappeared.  In reality they are absorbed by the different coordinates of the source and of the target evaluation point, which in this case, are numerically very different because of the use of two different projected coordinate reference systems for source and target.

The transformation parameter values (the coefficients) are not reversible.  For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction","For transformation Belge Lambert 72 to ED50 / UTM zone 31N,

Eo1 = 0
No1 = 0
Eo2 = 449681.702
No2 = 5460505.326
A1 = -71.3747
A2 = 1858.8407
A3 = -5.4504
A4 = -16.9681
A5 = 4.0783
A6 = 0.2193

For source coordinate system E1=200000  N1=100000, then
E2 = 647737.377  N2 = 5564124.227.",Coordinate pairs treated as complex numbers.  This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the regular 3rd-order polynomial.,EPSG guidance note #7.,OGP,2000/03/07,,1
9632,Complex polynomial (4th-order),0,"The relationship between two projected coordinate reference systems may be approximated more elegantly by a single polynomial regression formula written in terms of complex numbers. The advantage is that the dependence between the A and B coefficients (for U and V) is taken into account in the formula, resulting in fewer coefficients for the same order polynomial. A third-order polynomial in complex numbers is used in Belgium.  A fourth-order polynomial in complex numbers is used in The Netherlands for transforming coordinates referenced to the Amersfoort / RD system to and from ED50 / UTM.

(dX + i. dY) = (A1 + i. A2).(U + i.V) + (A3 + i. A4).(U + i.V)^2 + (A5 + i. A6).(U + i.V)^3 + (A7 + i.A8).(U + i.V)^4

where U = (XS - XS0).10-5
and     V = (YS - YS0).10-5

Then
XT  = XS - XS0 + XT0 + dX
YT  = YS - YS0 + YT0 + dY

where
XT , YT      are coordinates in the target coordinate reference system,
XS , YS      are coordinates in the source coordinate reference system,
XS0 , YS0   are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0   are coordinates of the evaluation point in the target coordinate reference system.

The transformation parameter values (the coefficients) are not reversible.  For the reverse transformation a different set of parameter values are required, used within the same formulas as the forward direction.","For transformation RD / Netherlands New to ED50 / UTM zone 31N,

Eo1 = 155000
No1 = 463000
Eo2 = 663395.607
No2 = 5781194.380
A1 = -51.681
A2 = 3290.525
A3 = 20.172
A4 = 1.133
A5 = 2.075
A6 = 0.251
A7 = 0.075
A8 = -0.012

For source coordinate system E1=200000  N1=500000, then
E2 =707155.557  N2 = 5819663.128.",Coordinate pairs treated as complex numbers.  This exploits the correlation between the polynomial coefficients and leads to a smaller number of coefficients than the regular 4th-order polynomial.,EPSG guidance note #7.,OGP,2000/03/07,,1
9633,Ordnance Survey National Transformation,1,"This method is used for transformation between ETRS89 (or WGS 84) geographic and OSGB36 / National Grid projected coordinate reference systems.  It uses a gridded data set with nodes in a temporary projected CRS (ETRS89 / National Grid) for bi-linear interpolation of grid coordinate differences.

The offsets at a point are derived by interpolation within the gridded data. One grid file contains both easting and northing offsets. The grid file format is given in documentation available from the information source. The coordinate reference system for the coordinates of the grid fiule nodes is an intermediary projected CRS, ETRS89 / British National Grid.

For the transformation of ETRS89 coordinates to OSGB36 / British National Grid coordinates, the ETRS89 latitude and longitude are first converted into ETRS89 / National Grid values using the Transverse Mercator formulas given in EPSG method code 9807. Then bi-linear interpolation is used to derive easting and northing offset values. These are added to the temporary ETRS89 / National Grid coordinate values to give OSGB36 / National Grid coordinate values.

For the transformation of OSGB36 / British National Grid coordinates to ETRS89 latitude and longitude, iteration is required. For the first iteration the OSGB36 / British National Grid easting and northing values are assumed to be in the ETRS89 / National Grid. Bi-linear interpolation is used to derive the easting and northing offset values which are applied with sign reversed to the input coordinates.The result is used in further iterations of interpolation and application of offset until the difference between results from successive iterations is insignificant. Finally these Transverse Mercator grid coordinates are converted to ETRS89 latitude and longitude using the formulae given in EPSG method code 9807.",See information source.,Geodetic transformation between ETRS89 (or WGS 84) and OSGB36 / National Grid.  Uses ETRS89 / National Grid as an intermediate coordinate system for bi-linear interpolation of gridded grid coordinate differences.,http://www.gps.gov.uk/gpssurveying.asp,OGP,2008/04/11,2004.190 2008.027,0
9634,Maritime Provinces polynomial interpolation,0,"The transformation makes use of a residual file for each Canadian maritime province.  The process of residual interpolation accounts for local variations in the coordinate reference system and provides a transformation accuracy of +/- 5 cm.

By using a second residual file, the transformation may be reversed.  Only one residual file is in use by the method during any given execution.",(none),This transformation is an executable module within the application NBGeocalc.  It is an adaptation of the ESTPM program developed by Geodetic Survey of Canada.,Survey of New Brunswick,OGP,2000/10/19,,0
9635,Geographic3D to Geographic2D+GravityRelatedHeight,1,"This is a complex, multi-step transformation, involving the application of a geoid height difference interpolated at a point in a ""geoid model"". The geoid model should be available as a regular grid of latitude and longitude with the height of the geoid above the ellipsoid at each grid node. Only the height is affected by this transformation; the geodetic latitude and longitude are not.

The transformation involves the following sequence of steps:
·  Selection of a subset of the geoid file covering the extent of the points to be transformed.
·  If the geoid file is not based on the source or target CRS, it needs to be transformed first. This involves transformation of the chosen subset of the geoid file from its orignal Geographic 3D CRS to the Geographic 3D CRS that is the source or the target of this transformation.
·  Calculation of the height of the geoid above the ellipsoid (""geoid undulation"") at the relevant point(s). This is achieved through a bi-linear interpolation of the geoid undulation, using the latitude and longitude to locate the point in the sub-grid.  This step results in the height of the geoid above the ellipsoid (N) of the Geographic 3D CRS, whether source or target.
·  At each point, the application of the calculated geoid undulation to the height to be transformed.

H=h-N for Geographic3D to Geographic2D+GravityRelatedHeight

h=H+N for Geographic2D+GravityRelatedHeight to Geographic3D

where h = the ellipsoidal height (height above the ellipsoid in a geographic 3D CRS)
and H = the gravity-related height (vertical CRS) component of the compound CRS.",(none),"Transformation from a Geographic 3D CRS to a Compound CRS consisting of a Geographic 2D CRS and a Vertical CRS, or vice versa. The Geographic 3D and the Geographic 2D CRS must be based on the same Geodetic Datum.","EPSG guidance note #7-2, http://www.epsg.org",OGP,2004/04/27,2004.190 2004.340,0
9636,Molodensky-Badekas (geog2D domain),1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

Transformation of coordinates from one geographic coordinate reference system into another is often carried out as a concatenation of the following operations:

(geographical to geocentric) + (geocentric to geocentric) + (geocentric to geographic)

The Coordinate Frame Rotation (geog2D domain) transformation has 5 steps:

(i) geographic 2D coordinates are converted to 3D using EPSG coordinate operation method code 9659;

(ii) geographic 3D coordinates are converted to geocentric coordinates using EPSG coordinate operation method code 9602;

(iii) the middle step of the concatenated transformation, from geocentric coordinates to geocentric coordinates, uses the Molodensky-Badekas (geocentric domain) method, EPSG method code 1034;

(iv) the geocentric coordinates are converted to geographic 3D using EPSG coordinate operation method code 9602;
 
(v) finally the geographic 3D coordinates are converted to geographic 2D using EPSG coordinate operation method code 9659.","Input point: 
Coordinate reference system: La Canoa (geographic 2D)
  Latitude =     9 deg 35 min 00.386 sec N
  Longitude = 66 deg 04 min 48.091 sec W
This is taken to be geographic 3D with an assumed Ellipsoidal height hS  =  201.465 m

This transforms to Cartesian geocentric coords:
   XS =  2 550 408.965 m
   YS = -5 749 912.266 m
   ZS =   1 054 891.114 m

Application of the 10 parameter Molodenski-Badekas Transformation results in REGVEN geocentric coordinates of:
   XT =  2 550 138.467 m
   YT = -5 749 799.862 m
   ZT  =  1 054 530.826 m

This converts into:
   Latitude =       9 deg 34 min 49.001 sec N
   Longitude =  66 deg 04 min 54.705 sec W
   Ellipsoidal height =  -18.10 m
on the REGVEN geographic 3D coordinate reference system.

Because the source coordinate reference system was 2D, the target system ellipsoidal height is ignored and the results treated as a geographic 2D coordinate reference system:
   Latitude =       9 deg 34 min 49.001 sec N
   Longitude =  66 deg 04 min 54.705 sec W",See methods 1034 and 1039 for similar tfms operating between other CRSs types.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2009/11/24,2002.510 2004.470 2009.083,0
9637,Degree representation conversion: deg to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

deg                     decimal degrees
adeg                   absolute value of decimal degrees
ideg                     integer degrees
sdeg                   signed integer degree
min                      real-number minutes
imin                      integer minutes
sec                      real-number seconds
lathem, lonhem    hemisphere abbreviation

Forward calculation from decimal degree representation to DMSH representation:
adeg = ABS(deg)
ideg = INT(adeg)

min = (adeg - ideg) * 60
imin = INT(min)
sec =  (min - imin) * 60

Then for latitude, if deg < 0, lathem = S else lathem = N
For longitude, if deg < 0, lonhem = W else lonhem = E

Reverse calculation from DMSH representation to decimal degree representation:
deg = (ideg + imin/60 + sec/3600) * H
where for latitude H = 1 if lathem = N and H = -1 if lathem = S
and for longitude  H = 1 if lonhem = E and H = -1 if lonhem = W","Source CRS = WGS 84 (deg) (CRS code 63266405).
Latitude = 35.75255, longitude = -85.20415

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9638,Degree representation conversion: degH to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

In this conversion (both forward and reverse) the hemisphere parameter remains unchanged and retains its position in the respective coordinate strings.

Forward calculation from degH representation to DMSH representation:
ideg = INT(adeg)

min = (adeg - ideg) * 60
imin = INT(min)
sec =  (min - imin) * 60

Reverse calculation from DMSH representation to decimal degree representation:
adeg = (ideg + imin/60 + sec/3600)","Source CRS = WGS 84 (degH) (CRS code 63266406).
Latitude = 35.75255N, longitude = 85.20415W

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9639,Degree representation conversion: Hdeg to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

In this conversion the hemisphere parameters retain their values but change their positions in their respective coordinate strings from the end of the strings to the beginnings (both forward and reverse).

Forward calculation from Hdeg representation to DMSH representation:
First, re-order fields from
          lathem, lat_adeg   and    lonhem, lon_adeg
to       lat_adeg, lathem   and    lon_adeg, lonhem

Then 
ideg = INT(adeg)

min = (adeg - ideg) * 60
imin = INT(min)
sec =  (min - imin) * 60

Reverse calculation from DMSH representation to Hdeg representation:
adeg = (ideg + imin/60 + sec/3600)

Then re-order fields from
          lat_adeg, lathem   and    lon_adeg, lonhem
to       lathem, lat_adeg   and    lonhem, lon_adeg","Source CRS = WGS 84 (Hdeg) (CRS code 63266407).
Latitude = N35.75255, longitude = W85.20415

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9640,Degree representation conversion: DM to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

Forward calculation from DM representation to DMSH representation:

ideg = ABS(sdeg)

imin = INT(min)
sec = (min - imin) * 60

If lat_sdeg < 0, lathem = S else lathem = N
If lon_sdeg < 0, lonhem = W else lathem = E

Reverse calculation from DMSH representation to DM representation:
sdeg = ideg * H
where for latitude, H = 1 if lathem = N and H = -1 if lathem = S
and for longitude  H = 1 if lonhem = E and H = -1 if lonhem = W

Then
min = imin + (sec / 60)","Source CRS = WGS 84 (DM) (CRS code 63266408).
Latitude = 35°45.153, longitude = -85°12.249

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9641,Degree representation conversion: DMH to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

The degree and hemisphere parameters remain unchanged in this conversion (both forward and reverse) and also retain their position in their respective coordinate strings.

Forward calculation from DMH representation to DMSH representation:
imin = INT(min)
sec = (min - imin) * 60

Reverse calculation from DMSH representation to DMH representation:
min = imin + (sec / 60)","Source CRS = WGS 84 (DMH) (CRS code 63266409).
Latitude = 35°45.153 N, longitude = 85°12.249 W

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9642,Degree representation conversion: HDM to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

In this conversion the degree parameters remain unchanged in this conversion (both forward and reverse). The hemisphere parameters retain their values but change their positions in their respective coordinate strings.

Forward calculation from HDM representation to DMSH representation:
imin = INT(min)
sec = (min - imin) * 60
Then reorder fields from hem, ideg, imin, sec to ideg, imin, sec, hem.

Reverse calculation from DMSH representation to HDM representation:
min = imin + (sec / 60)
Then re-order fields from ideg, min, hem to hem, ideg, min.","Source CRS = WGS 84 (HDM) (CRS code 63266410).
Latitude = N35°45.153, longitude =  W85°12.249

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9643,Degree representation conversion: DMS to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

In this conversion (both forward and reverse) the minute and second parameters remain unchanged.

Forward calculation from DMS representation to DMSH representation:

ideg = ABS(sdeg)
If lat_sdeg < 0, lathem = S else lathem = N
If lon_sdeg < 0, lonhem = W else lathem = E

Reverse calculation from DMSH representation to DMS representation:
sdeg = ideg * H
where for latitude, H = 1 if lathem = N and H = -1 if lathem = S
and for longitude  H = 1 if lonhem = E and H = -1 if lonhem = W","Source CRS = WGS 84 (DMS) (CRS code 63266411).
Latitude = 35°4509.18"", longitude = -85°1214.94""

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9644,Degree representation conversion: HDMS to DMSH,1,"In the formulas that follow, the coordinate strings are symbolically represented as follows:

In this conversion the parameter values remain unchanged but are re-ordered.

For the forward calculation from HDMS representation to DMSH representation, for each of latitude and longitude re-order the fields:
          from  	hem, ideg, imin, sec
          to       	ideg, imin, sec, hem

For the reverse calculation from DMSH representation to HDMS representation, for each of latitude and longitude re-order the fields:
          from  	ideg, imin, sec, hem 
          to       	hem, ideg, imin, sec","Source CRS = WGS 84 (HDMS) (CRS code 63266412).
Latitude = N35°4509.18"", longitude =  W85°1214.94""

Target CRS in DMSH = WGS 84 (CRS code 4326)
Latitude = 35°4509.18""N, longitude =  85°1214.94""W",Applies to 2D and the horizontal component of 3D ellipsoidal  systems.,EPSG guidance note #7.,OGP,2002/11/22,,1
9645,General polynomial of degree 2,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system to manageable numbers, between 10 and +10 at most.  This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.

Hence an evaluation point is chosen in the source coordinate reference system (XS0, YS0) and in the target coordinate reference system (XT0, YT0).  Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target coordinate reference system.  (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).

The selection of an evaluation point in each of the two coordinate reference systems allows the point coordinates in both to be reduced as follows:
XS - XS0 
YS - YS0
and
XT  XT0 
YT  YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding coordinate reference system. )

A further reduction step is usually necessary to bring these coordinate differences into the desired numerical range by applying a scaling factor to the coordinate differences in order to reduce them to a value range that may be applied to the polynomial formulae below without introducing numerical precision errors:

U = mS.(XS - XS0) 
V = mS.(YS - YS0)

where 
XS , YS  are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
mS is the scaling factor applied the coordinate differences in the source coordinate reference system.

The normalised coordinates U and V of the point whose coordinates are to be transformed are used as input to the polynomial transformation formula. In order to control the numerical range of the polynomial coefficients An and Bn the output coordinate differences dX and dY are multiplied by a scaling factor, mT.

mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2

mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2

from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.

The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A3 is represented as coordinate operation parameter Au2v0.

The relationship between the two coordinate reference systems can now be written as follows:

(XT - XT0) = (XS  XS0) + dX 
(YT - YT0) = (YS  YS0) + dY 
or
	XT = XS  XS0  + XT0 + dX 
YT = YS  YS0 + YT0 + dY

where:
XT , YT  are coordinates in the target coordinate reference system,
XS , YS  are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0  are coordinates of the evaluation point in the target coordinate reference system,
dX, dY   are derived through the scaled polynomial formulas.",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9646,General polynomial of degree 3,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

U = mS.(XS - XS0) 
V = mS.(YS - YS0)

mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3

mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3

from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.

The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A7 is represented as coordinate operation parameter Au2v1.

The relationship between the two coordinate reference systems can now be written as follows:

(XT - XT0) = (XS  XS0) + dX 
(YT - YT0) = (YS  YS0) + dY 
or
	XT = XS  XS0  + XT0 + dX 
YT = YS  YS0 + YT0 + dY

where:
XT , YT  are coordinates in the target coordinate reference system,
XS , YS  are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0  are coordinates of the evaluation point in the target coordinate reference system,
dX, dY   are derived through the scaled polynomial formulas.",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9647,General polynomial of degree 4,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

U = mS.(XS - XS0) 
V = mS.(YS - YS0)

mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
            + A10.U^4 + A11.U^3.V + A12.U^2.V^2 + A13.U.V^3 + A14.V^4

mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
            + B10.U^4 + B11.U^3.V + B12.U^2.V^2 + B13.U.V^3 + B14.V^4

from which dX and dY are evaluated. These will be in the units of the target coordinate reference system.

The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A13 is represented as coordinate operation parameter Au1v3.

The relationship between the two coordinate reference systems can now be written as follows:

(XT - XT0) = (XS  XS0) + dX 
(YT - YT0) = (YS  YS0) + dY 
or
	XT = XS  XS0  + XT0 + dX 
YT = YS  YS0 + YT0 + dY

where:
XT , YT  are coordinates in the target coordinate reference system,
XS , YS  are coordinates in the source coordinate reference system,
XS0 , YS0 are coordinates of the evaluation point in the source coordinate reference system,
XT0 , YT0  are coordinates of the evaluation point in the target coordinate reference system,
dX, dY   are derived through the scaled polynomial formulas.",(none),,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9648,General polynomial of degree 6,0,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as limitations in the transcription will be avoided.

The simplest of all polynomials is the general polynomial function. In order to avoid problems of numerical instability this type of polynomial should be used after reducing the coordinate values in both the source and the target coordinate reference system (CRS) to manageable numbers, between 10 and +10 at most.  This is achieved by working with offsets relative to a central evaluation point, scaled to the desired number range by applying a scaling factor to the coordinate offsets.

Hence an evaluation point is chosen in the source CRS (XS0, YS0) and in the target CRS (XT0, YT0).  Often these two sets of coordinates do not refer to the same physical point but two points are chosen that have the same coordinate values in both the source and the target CRS.  (When the two points have identical coordinates, these parameters are conveniently eliminated from the formulas, but the general case where the coordinates differ is given here).

The selection of an evaluation point in each of the two CRSs allows the point coordinates in both to be reduced as follows:
XS - XS0
YS - YS0
and
XT  XT0
YT  YT0
These coordinate differences are expressed in their own unit of measure, which may not be the same as that of the corresponding CRS.)

U = mS.(XS - XS0)
V = mS.(YS - YS0)

where
XS , YS  are coordinates in the source CRS,
XS0 , YS0 are coordinates of the evaluation point in the source CRS,
mS is the scaling factor applied the coordinate differences in the source CRS.

mT.dX = A0 + A1.U + A2.V + A3.U^2 + A4.U.V + A5.V^2 + A6.U^3 + A7.U^2.V + A8.U.V^2 + A9.V^3
            + A10.U^4 + A11.U^3.V + A12.U^2.V^2 + A13.U.V^3 + A14.V^4
            + A15.U^5 + A16.U^4.V + A17.U^3.V^2 + A18.U^2.V^3 + A19.U.V^4 + A20.V^5
            + A21.U^6 + A22.U^5.V + A23.U^4.V^2 + A24.U^3.V^3 + A25.U^2.V^4 + A26.U.V^5 + A27.V^6

mT.dY = B0 + B1.U + B2.V + B3.U^2 + B4.U.V + B5.V^2 + B6.U^3 + B7.U^2.V + B8.U.V^2 + B9.V^3
            + B10.U^4 + B11.U^3.V + B12.U^2.V^2 + B13.U.V^3 + B14.V^4
            + B15.U^5 + B16.U^4.V + B17.U^3.V^2 + B18.U^2.V^3 + B19.U.V^4 + B20.V^5
            + B21.U^6 + B22.U^5.V + B23.U^4.V^2 + B24.U^3.V^3 + B25.U^2.V^4 + B26.U.V^5 + B27.V^6

from which dX and dY are evaluated. These will be in the units of the target CRS.

The polynomial coefficients are given as parameters of the form Aumvn and Bumvn, where m is the power to which U is raised and n is the power to which V is raised. For example, A17 is represented as coordinate operation parameter Au3v2.

The relationship between the two CRSs can now be written as follows:

(XT - XT0) = (XS  XS0) + d
(YT - YT0) = (YS  YS0) + dY
or
	XT = XS  XS0  + XT0 + d
YT = YS  YS0 + YT0 + dY

where:
XT, YT are coordinates in the target CRS,
XS, YS are coordinates in the source CRS,
XS0, YS0 are coordinates of the evaluation point in the source CRS,
XT , YT0 are coordinates of the evaluation point in the target CRS,
dX, dY are derived through the scaled polynomial formulas.",See EPSG Guidance Note 7-2.,,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9649,Reversible polynomial of degree 2,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

See method code 9645 for description of general polynomial formula.

A general polynomial transformation is reversible only when the following conditions are met.
1. The co-ordinates of source and target evaluation point are (numerically) the same.
2. The unit of measure of the coordinate differences in source and target coordinate reference system are the same.
3. The scaling factors applied to source and target coordinate differences are the same.
4. The spatial variation of the differences between the coordinate reference systems around any given location is sufficiently small.

Clarification on conditions for polynomial reversibility:
Regarding 1 and 2 - 	In the reverse transformation the roles of the source and target coordinate reference systems are reversed.  Consequently, the co-ordinates of the evaluation point in the source coordinate reference system become those in the target coordinate reference system in the reverse transformation. Usage of the same transformation parameters for the reverse transformation will therefore only be valid if the evaluation point coordinates are numerically the same in source and target coordinate reference system and in the same units.  That is, XS0 = XT0 = X0 and YS0 = YT0 = Y0.
Re 3 - The same holds for the scaling factors of the source and target coordinate differences and for the units of measure of the coordinate differences. That is, mS = mT = m.
Re 4 - 	If conditions 1, 2 and 3 are all satisfied it then may be possible to use the forward polynomial algorithm with the forward parameters for the reverse transformation. This is the case if the spatial variations in dX and dY around any given location are sufficiently constant.  The signs of the polynomial coefficients are then reversed but the scaling factor and the evaluation point coordinates retain their signs. If these spatial variations in dX and dY are too large, for the reverse transformation iteration would be necessary.   It is therefore not the algorithm that determines whether a single step solution is sufficient or whether iteration is required, but the desired accuracy combined with the degree of spatial variability of dX and dY.

An example of a reversible polynomial is transformation is ED50 to ED87 (1) for the North Sea.  The suitability of this transformation to be described by a reversible polynomial can easily be explained. In the first place both source and target coordinate reference systems are of type geographic 2D. The typical difference in coordinate values between ED50 and ED87 is in the order of 2 metres (approximately 10E-6 degrees) in the area of application. The polynomial functions are evaluated about central points with coordinates of 55 deg N, 0 deg E in both coordinate reference systems. The reduced coordinate differences (in degrees) are used as input parameters to the polynomial functions. The output coordinate differences are corrections to the input coordinate offsets of about 10E-6 degrees. This difference of several orders of magnitude between input and output values is the property that makes a polynomial function reversible in practice (although not in a formal mathematical sense).

The error made by the polynomial approximation formulas in calculating the reverse correction is of the same order of magnitude as the ratio of output versus input:
(output error / input error) = (	output valu/ input value) which is approximately 10E-6

As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","See Reversible polynomial of degree 4, code 9651, for general methodology.",Reversibility is subject to constraints.  See Guidance Note 7 for conditions and clarification.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9650,Reversible polynomial of degree 3,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

See method code 9646 for description of general polynomial formula.

As long as the input values (the coordinate offsets from the evaluation point) are orders of magnitude larger than the output (the corrections), and provided the coefficients are used with changed signs, the polynomial transformation may be considered to be reversible.","See Reversible polynomial of degree 4, code 9651, for general methodology.",Reversibility is subject to constraints.  See Guidance Note 7 for conditions and clarification.,"EPSG guidance note #7-2, http://www.epsg.org",OGP,2002/12/21,,0
9651,Reversible polynomial of degree 4,1,"Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

See method code 9647 for description of general polynomial formula.