EPSG Dataset :: v12.004

Oblique Stereographic

Coordinate Operation Method Details [VALID]

Name:

Oblique Stereographic

Code:

9809

Operation is Reversible:

Yes

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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
Latitude of natural origin	8801	No	The latitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the latitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0).
Longitude of natural origin	8802	No	The longitude of the point from which the values of both the geographical coordinates on the ellipsoid and the grid coordinates on the projection are deemed to increment or decrement for computational purposes. Alternatively it may be considered as the longitude of the point which in the absence of application of false coordinates has grid coordinates of (0,0). Sometimes known as "central meridian (CM)".
Scale factor at natural origin	8805	No	The factor by which the map grid is reduced or enlarged during the projection process, defined by its value at the natural origin.
False easting	8806	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Easting, FE, is the value assigned to the abscissa (east or west) axis of the projection grid at the natural origin.
False northing	8807	No	Since the natural origin may be at or near the centre of the projection and under normal coordinate circumstances would thus give rise to negative coordinates over parts of the mapped area, this origin is usually given false coordinates which are large enough to avoid this inconvenience. The False Northing, FN, is the value assigned to the ordinate (north or south) axis of the projection grid at the natural origin.

Meta Data

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.

Information Source:

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

Data Source:

EPSG

Revision Date:

December 3, 2021

Change ID:

[1999.811] [2006.2] [2017.018] [2017.024] [2021.105]

Alias:

Alias	Naming System	Remarks
Roussilhe	EPSG alias

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.