# EPSG:9810

## Polar Stereographic (variant A)

### Attributes

Data source: EPSG

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

Revision date: 2018-08-29

Remarks: Latitude of natural origin must be either 90 degrees or -90 degrees (or equivalent in alternative angle unit).

### 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.

For the forward conversion from latitude and longitude, for the south pole case

E = FE + rho * sin(lon – lonO)
N = FN + rho * cos(lon – lonO)
where
t = tan(pi/4 + lat/2) / {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)}
rho = 2*a*ko*t / {[(1+e)^(1+e) (1–e)^(1–e)]^0.5}

For the north pole case,
rho and E are found as for the south pole case but
t  = tan(pi/4 – lat/2) *  {[(1 + e sin(lat)) / (1 – e sin(lat))]^(e/2)}
N = FN – rho * cos(lon – lonO)

For the reverse conversion from easting and northing to latitude and longitude,
lat = chi + (e^2/2 + 5e^4/24 + e^6/12 + 13e^8/360) sin(2 chi)
+ (7e^4/48 + 29e^6/240 + 811e^8/11520) sin(4 chi)
+ (7e^6/120 +  81e^8/1120) sin(6 chi)  + (4279e^8/161280) sin(8 chi)

where rho'  = [(E-FE)^2  + (N – FN)^2]^0.5
t'   =rho' {[(1+e)^(1+e) * (1– e)^(1-e)]^0.5} / (2 a ko)
and for the south pole case
chi  = 2 atan(t' ) – pi/2
but for the north pole case
chi  =  pi/2 - 2 atan(t')

Then for for both north and south cases if E = FE, lon = lonO
else for the south pole case
lon = lonO + atan2[(E – FE),(N – FN)]
and for the north pole case
lon = lonO + atan2[(E – FE),(FN – N)] (see GN7-2 implementation notes in preface for atan2 convention)```

### Example

```For Projected Coordinate Reference System: WGS 84 / UPS North

Parameters:
Ellipsoid: WGS 84
a = 6378137.0 metre
1/f = 298.2572236
then e = 0.081819191

Latitude of natural origin (latO): 90°00'00.000"N =1.570796327 rad
Longitude of origin (longO): 0°00'00.000"E=0.0 rad
Scale factor at natural origin (ko): 0.994
False easting (FE) 2000000.00 metre
False northing (FN) 2000000.00 metre

Forward calculation for:

t  = 0.150412808
rho = 1900814.564
whence
E = 3320416.75 m
N =  632668.43 m

Reverse calculation for the same Easting and Northing (3320416.75 E, 632668.43 N) first gives:
rho' = 1900814.566
t'  = 0.150412808
chi  = 1.2722090

Then
Latitude (lat) = 73°00'00.000"N
Longitude (lon) = 44°00'00.000"E```