Lambert Conformal Conic Projection Calculator
This calculator applies the Lambert conformal conic projection with two standard parallels in the forward direction: it takes the latitude and longitude of a point and returns the projected coordinates x and y together with the point scale factor k. It works on an ellipsoid or on a sphere and follows J. P. Snyder, Map Projections - A Working Manual, USGS Professional Paper 1395, 1987, equations 15-1 to 15-10.
It is meant for surveyors, GIS analysts and cartographers who already hold a projection definition: the semi-major axis a, the eccentricity squared e², the two standard parallels, and the origin latitude and longitude. Every intermediate quantity (m1, m2, t1, t2, t0, n, F, ρ0, t, ρ and θ) is shown, and the formulas are written out with the values you typed. The fields start with the ellipsoid example from Snyder, pages 296-297.
Forward calculation
Projected point
Intermediate values
| Quantity | Value | Unit |
|---|---|---|
| m1, at φ1 | — | dimensionless |
| m2, at φ2 | — | dimensionless |
| t1, at φ1 | — | dimensionless |
| t2, at φ2 | — | dimensionless |
| t0, at φ0 | — | dimensionless |
| n | — | dimensionless |
| F | — | dimensionless |
| ρ0 | — | unit of a |
| t, at φ | — | dimensionless |
| ρ | — | unit of a |
| θ | — | degrees |
Formulas with your values
- —
All values are carried in double precision with no rounding inside the calculation; only the displayed results are rounded to 7 decimals.
How to use the calculator
- Enter the semi-major axis a in any length unit (above 0) and the eccentricity squared e² (0 or more and below 1). For a sphere, enter e² = 0; a is then its radius.
- Enter the two standard parallels φ1 and φ2 in degrees. Each must be from 0 to 89 degrees, and φ1 must not equal φ2.
- Enter the origin latitude φ0 (0 to 89 degrees) and the origin longitude λ0 (−180 to 180 degrees, west negative).
- Enter the point latitude φ (0 to 89 degrees) and the point longitude λ (−180 to 180 degrees, west negative). The difference λ − λ0 must be from −180 to 180 degrees; it is not wrapped.
- Results update as you type; the Calculate button runs the same calculation. Read x, y and k, then check the intermediate values and the formulas written out with your values. A field that is empty or out of range is named in a message below it, and the results show dashes.
Method
Snyder, equations 15-1 to 15-10, forward direction, with degrees converted to radians as π/180:
- e = √e²
- m(φ) = cos φ / √(1 − e² sin² φ)
- t(φ) = tan(π/4 − φ/2) / [(1 − e sin φ) / (1 + e sin φ)]^(e/2)
- m1 = m(φ1), m2 = m(φ2), t1 = t(φ1), t2 = t(φ2), t0 = t(φ0), t = t(φ)
- n = (ln m1 − ln m2) / (ln t1 − ln t2)
- F = m1 / (n t1^n)
- ρ0 = a F t0^n
- ρ = a F t^n
- θ = n (λ − λ0)
- x = ρ sin θ
- y = ρ0 − ρ cos θ
- k = ρ n / (a m(φ))
Frequently asked questions
Why does the calculator ask for e² and not for e?
The eccentricity squared e² is the only ellipsoid input, and e is computed from it as the square root of e². Entering a rounded e instead can shift ρ by 0.07 m.
How do I compute on a sphere instead of an ellipsoid?
Enter 0 for the eccentricity squared e². The semi-major axis a is then the radius of the sphere. With a = 1 and the angles of Snyder's example, the calculator gives n = 0.6304777, x = 0.2966785 and y = 0.2462112.
What units are x, y and k in?
x, y, ρ0 and ρ are in the length unit you used for the semi-major axis a. The point scale factor k is dimensionless and equals 1 on both standard parallels. θ is shown in degrees.
Does the calculator convert GPS or WGS84 coordinates?
No. It is a calculation aid for the projection as defined and performs no datum transformation. A GPS latitude is not converted to the ellipsoid you enter.
Note
This page is a calculation aid for the Lambert conformal conic projection as defined by the values you enter. It performs 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.