Polar Azimuthal Equidistant Projection Calculator
This calculator applies the forward equations of the azimuthal equidistant projection in its north polar aspect on the ellipsoid. It takes the latitude and longitude of a point and returns the rectangular coordinates x and y, the radius ρ from the pole, and the scale factor k along the parallel, following J. P. Snyder, USGS Professional Paper 1395 (1987), equations 21-30 to 21-32, 25-16 and 3-21. The North Pole is the centre of the map.
It is written for surveyors, GIS analysts and cartographers. The ellipsoid is entered as the semi-major axis a and the eccentricity squared e²; e⁴ and e⁶ are derived from e², degrees are converted to radians with π/180, and every step runs in double precision with no rounding inside. Lengths are shown to 4 decimals in the unit of a, and m and k to 7 decimals. The fields start with Snyder's worked example on page 339.
Forward calculation: latitude and longitude to x, y and k
Results update as you type.
Results
- x (unit of a)
- —
- y (unit of a)
- —
- k, scale factor along the parallel (dimensionless)
- —
- ρ = Mp − M (unit of a)
- —
- M at φ (unit of a)
- —
- Mp = M(π/2) (unit of a)
- —
- m (dimensionless)
- —
Formulas with your values
Intermediate values inside these lines carry two more decimals than the results.
- λ − λ₀ —
- M = a [ (1 − e²/4 − 3e⁴/64 − 5e⁶/256) φ − (3e²/8 + 3e⁴/32 + 45e⁶/1024) sin 2φ + (15e⁴/256 + 45e⁶/1024) sin 4φ − (35e⁶/3072) sin 6φ ], with e⁴ = (e²)², e⁶ = (e²)³ and φ in radians —
- Mp = M(π/2) —
- ρ = Mp − M —
- m = cos φ / √(1 − e² sin² φ) —
- x = ρ sin(λ − λ₀) —
- y = −ρ cos(λ − λ₀) —
- k = ρ / (a m); at φ = 90° no division: x = 0, y = 0, k = 1 —
How to use the calculator
- Enter the semi-major axis a in any length unit, above 0. The lengths x, y, ρ, M and Mp are returned in that same unit.
- Enter the eccentricity squared e², dimensionless, above 0 and below 1.
- Enter the central meridian λ₀ in degrees, west negative, from −180 to 180.
- Enter the point latitude φ in degrees, from 0 to 90 (north only).
- Enter the point longitude λ in degrees, west negative, from −180 to 180. The difference λ − λ₀ must also lie from −180 to 180; it is not wrapped.
- Read x, y, k, ρ, M, Mp and m. Results recompute as you type, and the Calculate button recomputes on demand. An empty or invalid field shows dashes and a message naming that field.
Frequently asked questions
Which map aspect does this calculator use?
Only the north polar aspect, with the North Pole at the centre of the map. Latitudes from 0 to 90 degrees north are accepted; the south polar, oblique and equatorial aspects are not offered.
What is the scale factor k?
k is the scale factor along the parallel, computed as k = ρ / (a m) with m = cos φ / √(1 − e² sin² φ). At the pole (φ = 90°) there is no division: x = 0, y = 0 and k = 1.
Why does the calculator ask for e² and not e?
The eccentricity squared e² is the only ellipsoid input besides a: the series for M uses e², e⁴ = (e²)² and e⁶ = (e²)³, so e itself is never needed. e² must be above 0 and below 1; a sphere is not offered.
Does the calculator convert GPS coordinates to my ellipsoid?
No. It is a calculation aid for the projection as defined, with no datum transformation: a GPS latitude is not converted to the ellipsoid entered. Enter latitude and longitude already referred to the ellipsoid given by a and e².
Note
This page covers the north polar aspect only, with the pole at the centre of the map. It is a calculation aid for the projection as defined, 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.