Oblique Stereographic Projection Calculator
This calculator projects a point given by latitude φ and longitude λ onto the oblique stereographic projection of the ellipsoid and returns the plane coordinates x and y and the point scale factor k. It follows the forward equations 21-24 to 21-27 of J. P. Snyder, Map Projections - A Working Manual (USGS Professional Paper 1395, 1987), for surveyors, GIS analysts and cartographers who need to check or reproduce these values.
The ellipsoid is defined by its semi-major axis a, in any length unit, and its eccentricity squared e². The projection is centred at latitude φ1 and longitude λ0, with scale factor k0 at the centre. Alongside x, y and k the page shows the quantities m1, m and A, and writes out each formula with the values you entered so that every step can be followed.
Forward calculation: latitude and longitude to x, y and k
Ellipsoid and projection
Point to project
Formulas with your values
Angles are converted from degrees to radians by multiplying by π/180. Intermediate values are kept at full double precision; only the final results are rounded.
e = √(e²)
χ(p) = 2 · atan(tan(π/4 + p/2) · ((1 − e · sin p)/(1 + e · sin p))e/2) − π/2
χ1 = χ(φ1), χ = χ(φ)
m1 = cos φ1 / √(1 − e² · sin² φ1)
m = cos φ / √(1 − e² · sin² φ)
den = 1 + sin χ1 · sin χ + cos χ1 · cos χ · cos(λ − λ0)
A = 2 · a · k0 · m1 / (cos χ1 · den)
x = A · cos χ · sin(λ − λ0)
y = A · (cos χ1 · sin χ − sin χ1 · cos χ · cos(λ − λ0))
k = A · cos χ / (a · m)
How to use the calculator
- Enter the semi-major axis a of the ellipsoid in any length unit, and its eccentricity squared e² (dimensionless, above 0 and below 1).
- Enter the scale factor at the centre k0 (dimensionless, above 0).
- Enter the centre of the projection: latitude φ1 from 1 to 89 degrees north and longitude λ0 from −180 to 180 degrees, west negative.
- Enter the point to project: latitude φ from −89 to 89 degrees and longitude λ from −180 to 180 degrees, west negative, keeping λ − λ0 between −180 and 180 degrees.
- Read x and y in the unit of a and the point scale factor k, together with m1, m and A. Results update as you type; the Calculate button recomputes them, and the formulas show the values you entered.
Frequently asked questions
Is this the same as the EPSG 9809 Oblique Stereographic method?
No. This calculator uses the oblique ellipsoidal stereographic projection as given by Snyder in USGS Professional Paper 1395. The EPSG 9809 method (Oblique Stereographic, double stereographic) is constructed differently.
In which units are x, y and A given?
In the unit of the semi-major axis a. Enter a in metres and x, y and A are in metres; enter it in feet and they are in feet. The point scale factor k and the quantities m1 and m are dimensionless.
Does the calculator convert GPS coordinates to my ellipsoid?
No. It is a calculation aid with no datum transformation: a GPS latitude is not converted to the ellipsoid entered. The latitude and longitude you enter are used as they are on the ellipsoid defined by a and e².
Which input ranges are accepted?
The semi-major axis a and the scale factor k0 must be above 0, and e² above 0 and below 1. The centre latitude φ1 must be from 1 to 89 degrees (north only) and the point latitude φ from −89 to 89 degrees. Both longitudes must be from −180 to 180 degrees, west negative, and λ − λ0 must also be from −180 to 180 degrees, with no wrapping. The antipode of the centre cannot be projected.
This is the oblique ellipsoidal stereographic projection as in Snyder, not the EPSG 9809 method (Oblique Stereographic, double stereographic), which is constructed differently. Only the forward direction is computed.
This page is a calculation aid with no datum transformation: a GPS latitude is not converted to the ellipsoid entered.
Source: J. P. Snyder, Map Projections - A Working Manual, USGS Professional Paper 1395, 1987, equations 21-24 to 21-27.