EPSG:9809

Oblique Stereographic

Attributes

Data source: EPSG

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2021-12-03

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.

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(lat0) / (1 - e^2)]}^0.5
c=  [(n+sin(lat0)) (1-sin(chi00))]/[(n-sin(lat0)) (1+sin(chi00))]

where:
sin(chi00) = (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 of the origin (chi0) is then computed from:

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

where w2 = c w1
 
For any point with geodetic latitude (lat) the equivalent conformal latitude (chi) is then computed from: 
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
dLambda = Lambda - Lamda0 = n(lon-lon0) where Lambda is the conformal longitude at geodetic longitude (lon)
B = [1+sin(chi) sin(chi0) + cos(chi) cos(chi0) cos(dLambda)]

and
E = FE + 2 R k0 cos(chi) sin(Lambda-Lambda0) / B
N = FN + 2 R k0 [sin(chi) cos(chi0) - cos(chi) sin(chi0) cos(dLambda)] / 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 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)]

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 for atan2 convention)

Geodetic longitude lon = (Lambda-Lambda0 ) / n +  Lambda0

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

The first approximation for geodetic latitude is found from:

lat1 = 2 atan(e^psi)  - pi/2  

where e=base of natural logarithms
            psi = isometric latitude at lat1
                   = ln[{tan(lat1 / 2 + pi/4}  {(1-e sin(lat1))/(1+e sin(lat1))}^(e/2)]
 
Then iterate: 
psi1 = ln[{tan(lati/2 + pi/4}  {(1-e sin(lati))/(1+e sin(lati))}^(e/2)]
and
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 and lon must be reversed to be used in the equations and lat as a southerly latitude will be negative anyway.

If the projection is the equatorial case, lat0 and chi0 will be zero degrees and the formulas may be simplified as a result, but the above formulas remain valid.

For the polar case, lat0 and chi0 will be 90 degrees and the formulas become indeterminate. See polar steroegraphic method for formulas for the polar case.

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.

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 = 6374588.710    nu0 = 6390710.613    R = 6382644.571
S1 = 8.509582274       S2 = 0.878790173      w1 = 8.428769183
sin chi00 = 0.787883237   n = 1.000475857    c  = 1.007576465
then
w2 = 8.492629457 

and the conformal coordinate of the map projection origin are: 
 chi0 = 0.909684757      Lamda0 = d0 

for the point  (lat, lon)
lat = 0.925024504 rad
Sa = 8.932237807       Sb = 0.8779500613      w = 8.913578649
chi  = 0.924394997    

lon = 0.104719755 rad    dLambda = 0.010692803    B = 1.999870665

Then    
E = 196105.283    N = 557057.739    

reverse calculation for the same Easting and Northing (196105.28, 557057.739) first gives:

g = 4379954.188     h = 37197327.960    i = 0.00110272    j = 0.008488259

then  
chi  = 0.924394767    psi = 1.089495505      lat1 = 0.921804948

Iteration for psi and phi:
                                       psi1 = 1.084170545    lat2 = 0.925031393       
                                       psi2 = 1.089506925    lat3 = 0.925024504       
                                       psi3 = 1.089495505    lat4 = 0.925024504 rad

Also  dLambda = 0.010692803 whence lon = 0.104719755 rad

Then Latitude      = 53°00'00.000"N
          Longitude   =   6°00'00.000"E
MapTiler banner