x (unit of a)—
y (unit of a)—
Lambert Azimuthal Equal-Area Projection Calculator (Oblique, Ellipsoid)
This calculator projects a point given by latitude and longitude to plane coordinates x and y with the Lambert azimuthal equal-area projection in its oblique aspect on the ellipsoid. It follows the forward equations 24-17 to 24-20 of J. P. Snyder, Map Projections - A Working Manual, USGS Professional Paper 1395, 1987, and is written for surveyors, GIS analysts and cartographers.
Enter the semi-major axis a and the eccentricity squared e² of the ellipsoid, the centre of the projection (φ1, λ0) and the point (φ, λ). Along with x and y the page shows every intermediate quantity — q, q1, qp, β, β1, Rq, m1, B and D — and the formulas with your values, so each step can be checked against Snyder's worked example on page 334, which prints x = −965,932.1 and y = −1,056,814.9 for a = 6378206.4, e² = 0.00676866, a centre at 40° N, 100° W and a point at 30° N, 110° W.
Inputs
Results
| q (dimensionless) | — |
|---|---|
| q1 (dimensionless) | — |
| qp (dimensionless) | — |
| β (degrees) | — |
| β1 (degrees) | — |
| Rq (unit of a) | — |
| m1 (dimensionless) | — |
| B (unit of a) | — |
| D (dimensionless) | — |
Formulas with your values
- e = √e²
- q(p) = (1 − e²)·[sin p/(1 − e²·sin² p) − ln((1 − e·sin p)/(1 + e·sin p))/(2e)]
- qp = q(90°), q1 = q(φ1), q = q(φ)
- β1 = asin(q1/qp), β = asin(q/qp)
- Rq = a·√(qp/2)
- m1 = cos φ1/√(1 − e²·sin² φ1)
- D = a·m1/(Rq·cos β1)
- den = 1 + sin β1·sin β + cos β1·cos β·cos(λ − λ0); if den is below 1e-12 the point is the antipode of the centre
- B = Rq·√(2/den)
- x = B·D·cos β·sin(λ − λ0)
- y = (B/D)·(cos β1·sin β − sin β1·cos β·cos(λ − λ0))
How to use the calculator
- Enter the semi-major axis a of the ellipsoid in any length unit, above 0. The lengths x, y, Rq and B are returned in the same unit.
- Enter the eccentricity squared e², above 0 and below 1. It is the only ellipsoid shape input; a sphere is not offered.
- Enter the centre latitude φ1 in degrees, from 1 to 89 north, and the centre longitude λ0 in degrees, from −180 to 180, west negative.
- Enter the point latitude φ in degrees, from −89 to 89, and the point longitude λ in degrees, from −180 to 180, west negative. The difference λ − λ0 must also lie from −180 to 180; no wrapping is applied.
- Read x and y and the intermediate quantities. Results update as you type; the Calculate button runs the same computation. Check each step in the formulas with your values.
Frequently asked questions
Which equations does this calculator use?
It uses the forward equations for the oblique aspect of the Lambert azimuthal equal-area projection on the ellipsoid, equations 24-17 to 24-20 in J. P. Snyder, Map Projections - A Working Manual, USGS Professional Paper 1395, 1987. All quantities are computed in double precision with no rounding inside the calculation.
In which unit are x and y given?
x, y, Rq and B are lengths in the same unit as the semi-major axis a you enter, shown to 4 decimals. q, q1, qp, m1 and D are dimensionless and β and β1 are in degrees, all shown to 7 decimals.
Does the calculator convert a GPS latitude to my ellipsoid?
No. This page is a calculation aid for the projection as defined, with no datum transformation. A GPS latitude is not converted to the ellipsoid entered, so enter a latitude and longitude that already refer to that ellipsoid.
Why is a longitude difference over 180 degrees rejected?
Longitudes are entered in degrees, west negative, from -180 to 180, and the difference between the point longitude and the centre longitude must also be from -180 to 180. No wrapping is applied, so a point longitude of 170 with a centre longitude of -170 is reported as an error on the point longitude field.