Oblique Stereographic

Definition: OGP XML

<div class="syntax"><pre><?xml version="1.0" encoding="UTF-8"?> <gml:OperationMethod xmlns:epsg="urn:x-ogp:spec:schema-xsd:EPSG:1.0:dataset" xmlns:gml="http://www.opengis.net/gml/3.2" xmlns:xlink="http://www.w3.org/1999/xlink" gml:id="iogp-method-9809"> <gml:metaDataProperty> <epsg:CommonMetaData> <epsg:alias alias="Roussilhe" code="418" codeSpace="urn:ogc:def:naming-system:EPSG::7301" /> <epsg:informationSource>EPSG guidance note #7-2, http://www.epsg.org</epsg:informationSource> <epsg:revisionDate>2018-08-29</epsg:revisionDate> <epsg:changes> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::1999.811" /> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2006.200" /> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2017.018" /> <epsg:changeID xlink:href="urn:ogc:def:change-request:EPSG::2017.024" /> </epsg:changes> <epsg:show>true</epsg:show> <epsg:isDeprecated>false</epsg:isDeprecated> </epsg:CommonMetaData> </gml:metaDataProperty> <gml:metaDataProperty> <epsg:CoordinateOperationMethodMetaData> <epsg:isOperationReversible>true</epsg:isOperationReversible> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8801" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8802" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8805" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8806" /> <epsg:signReversal changeSign="false" xlink:href="urn:ogc:def:parameter:EPSG::8807" /> <epsg:example>For Projected Coordinate System RD / Netherlands New Parameters: Ellipsoid Bessel 1841 a = 6377397.155 m 1/f = 299.15281 then e = 0.08169683 Latitude Natural Origin 52°09'22.178"N = 0.910296727 rad Longitude Natural Origin 5°23'15.500"E = 0.094032038 rad Scale factor k0 0.9999079 False Eastings FE 155000.00 m False Northings FN 463000.00 m Forward calculation for: Latitude 53°N = 0.925024504 rad Longitude 6°E = 0.104719755 rad first gives the conformal sphere constants: rho0 = 6374588.71 nu0 = 6390710.613 R = 6382644.571 n = 1.000475857 c = 1.007576465 where S1 = 8.509582274 S2 = 0.878790173 w1 = 8.428769183 sin chi0 = 0.787883237 w = 8.492629457 chi0 = 0.909684757 D0 = d0 for the point chi = 0.924394997 D = 0.104724841 hence B = 1.999870665 N = 557057.739 E = 196105.283 reverse calculation for the same Easting and Northing first gives: g = 4379954.188 h = 37197327.96 i = 0.001102255 j = 0.008488122 then D = 0.10472467 Longitude = 0.104719584 rad = 6 deg E chi = 0.924394767 psi = 1.089495123 phi1 = 0.921804948 psi1 = 1.084170164 phi2 = 0.925031162 psi2 = 1.089506925 phi3 = 0.925024504 psi3 = 1.089495505 phi4 = 0.925024504 Then Latitude = 53°00'00.000"N Longitude = 6°00'00.000"E</epsg:example> </epsg:CoordinateOperationMethodMetaData> </gml:metaDataProperty> <gml:identifier codeSpace="IOGP">urn:ogc:def:method:EPSG::9809</gml:identifier> <gml:name>Oblique Stereographic</gml:name> <gml:remarks>This is not the same as the projection method of the same name in USGS Professional Paper no. 1395, "Map Projections - A Working Manual" by John P. Snyder.</gml:remarks> <gml: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. Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are: R= sqrt( rho0 * nu0) n= {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5 c= [(n+sin(lat0)) (1-sin(chi0))]/[(n-sin(lat0)) (1+sin(chi0))] where: sin(chi0) = (w1-1)/(w1+1) w1 = (S1.(S2)^e)^n S1 = (1+sin(lat0))/(1-sin(lat0)) S2 = (1-e sin(lat0))/(1+e sin(lat0)) The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from : chi0 = asin[(w2-1)/(w2+1)] where S1 and S2 are as above and w2 = c (S1(S2)^e)^n lambda0 = lon0 For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from lambda = n(lon-lambda0) + lambda0 chi = asin[(w-1)/(w+1)] where w = c (Sa (Sb)^e)^n Sa = (1+sin(lat))/(1-sin(lat)) Sb = (1-e.sin(lat))/(1+e.sin(lat)) Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)] N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values. The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) : chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)] lambda = j + 2 i + lambda0 where g = 2 R k0 tan(pi/4 - chi0/2) h = 4 R k0 tan(chi0) + g i = atan2[(E-FE) , {h+(N-FN)}] j = atan2[(E-FE) , (g-(N-FN)] - i (see GN7-2 implementation notes in preface for atan2 convention) Geodetic longitude lon = (lambda-lambda0 ) / n + lambda0 Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1- sin(chi))}] / n First approximation lat1 = 2 atan(e^psi) - pi/2 where e=base of natural logarithms. psii = isometric latitude at lati where psii= ln[{tan(lati/2 + pi/4} {(1-e sin(lati))/(1+e sin(lati))}^(e/2)] Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2) until the change in lat is sufficiently small. For Oblique Stereographic projections centred on points in the southern hemisphere, the signs of E, N, lon0, lon, must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude. An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed. The conformal longitude is then always equivalent to the geodetic longitude. This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different coordinate operation method to that described above.</gml:formula> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8801" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8802" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8805" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8806" /> <gml:generalOperationParameter xlink:href="urn:ogc:def:parameter:EPSG::8807" /> </gml:OperationMethod> </pre></div>

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      <epsg:example>For Projected Coordinate System RD / Netherlands New

Parameters:
Ellipsoid   Bessel 1841    a = 6377397.155 m    1/f = 299.15281
then e = 0.08169683

Latitude Natural Origin      52°09'22.178"N  = 0.910296727 rad
Longitude Natural Origin     5°23'15.500"E  =  0.094032038 rad
Scale factor k0                 0.9999079
False Eastings FE             155000.00 m
False Northings FN           463000.00 m

Forward calculation for: 

Latitude    53°N = 0.925024504 rad
Longitude   6°E = 0.104719755 rad

first gives the conformal sphere constants:

rho0 = 6374588.71    nu0 = 6390710.613
R = 6382644.571    n = 1.000475857    c  = 1.007576465

where S1 = 8.509582274  S2 = 0.878790173  w1 = 8.428769183
sin chi0 = 0.787883237

w   = 8.492629457   chi0 = 0.909684757      D0 = d0 

for the point  chi  = 0.924394997    D = 0.104724841

hence B = 1.999870665    N = 557057.739    E = 196105.283

reverse calculation for the same Easting and Northing first gives:

g = 4379954.188    h = 37197327.96   i = 0.001102255   j = 0.008488122

then  D = 0.10472467  Longitude = 0.104719584 rad =  6 deg E

chi  = 0.924394767    psi = 1.089495123
phi1 = 0.921804948       psi1 = 1.084170164
phi2 = 0.925031162       psi2 = 1.089506925
phi3 = 0.925024504       psi3 = 1.089495505
phi4 = 0.925024504

Then Latitude      = 53°00'00.000"N
          Longitude   =   6°00'00.000"E</epsg:example>
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  <gml:identifier codeSpace="IOGP">urn:ogc:def:method:EPSG::9809</gml:identifier>
  <gml:name>Oblique Stereographic</gml:name>
  <gml:remarks>This is not the same as the projection method of the same name in USGS Professional Paper no. 1395, "Map Projections - A Working Manual" by John P. Snyder.</gml:remarks>
  <gml: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.

Given the geodetic origin of the projection at the tangent point (lat0, lon0), the parameters defining the conformal sphere are:

R= sqrt( rho0 * nu0)
n= {1 + [e^2 * cos^4(latC) / (1 - e^2)]}^0.5
c=  [(n+sin(lat0)) (1-sin(chi0))]/[(n-sin(lat0)) (1+sin(chi0))]

where:
sin(chi0) = (w1-1)/(w1+1)
w1 = (S1.(S2)^e)^n
S1 = (1+sin(lat0))/(1-sin(lat0))
S2 = (1-e sin(lat0))/(1+e sin(lat0))

The conformal latitude and longitude (chi0,lambda0) of the origin are then computed from :

chi0 = asin[(w2-1)/(w2+1)]

where S1 and S2 are as above and  w2 = c (S1(S2)^e)^n
 
lambda0  = lon0

For any point with geodetic coordinates (lat, lon) the equivalent conformal latitude and longitude (chi, lambda) are computed from 
lambda = n(lon-lambda0) + lambda0
chi = asin[(w-1)/(w+1)]

where w = c (Sa (Sb)^e)^n
Sa = (1+sin(lat))/(1-sin(lat))
Sb = (1-e.sin(lat))/(1+e.sin(lat))
 
Then B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(lambda-lambda0)]

N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(lambda-lambda0)] / B

E = FE + 2 R k0 cos(chi) sin(lambda-lambda0) / B


The reverse formulae to compute the geodetic coordinates from the grid coordinates involves computing the conformal values, then the isometric latitude and finally the geodetic values.

The parameters of the conformal sphere and conformal latitude and longitude at the origin are computed as above. Then for any point with Stereographic grid coordinates (E,N) :

chi = chi0 + 2 atan[{(N-FN)-(E-FE) tan (j/2)} / (2 R k0)]

lambda = j + 2 i + lambda0

where g = 2 R k0 tan(pi/4 - chi0/2)
h = 4 R k0 tan(chi0) + g
i = atan2[(E-FE) , {h+(N-FN)}]
j = atan2[(E-FE) , (g-(N-FN)] - i
(see GN7-2 implementation notes in preface for atan2 convention)

Geodetic longitude lon = (lambda-lambda0 ) / n +  lambda0

Isometric latitude psi = 0.5 ln [(1+ sin(chi)) / { c (1-  sin(chi))}] / n

First approximation lat1 = 2 atan(e^psi)  - pi/2  where e=base of natural logarithms.

psii = isometric latitude at lati

where psii= ln[{tan(lati/2 + pi/4}  {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]
 
Then iterate lat(i+1) = lati - ( psii - psi ) cos(lati) (1 -e^2 sin^2(lati)) / (1 - e^2)

until the change in lat is sufficiently small.

For Oblique Stereographic projections centred on points in the southern hemisphere,  the signs of E, N, lon0, lon,  must be reversed to be used in the equations and lat will be negative anyway as a southerly latitude.

An alternative approach is given by Snyder, where, instead of defining a single conformal sphere at the origin point, the conformal latitude at each point on the ellipsoid is computed.  The conformal longitude is then always equivalent to the geodetic longitude.  This approach is a valid alternative to the above, but gives slightly different results away from the origin point. It is therefore considered by EPSG to be a different coordinate operation method to that described above.</gml:formula>
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EPSG:9809

Oblique Stereographic

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