Map Projection Calculator
Every flat map is a mathematical compromise: projecting the curved Earth onto a plane always distorts something. This calculator performs the forward projection of a latitude/longitude point to x/y coordinates in metres, using the spherical forms of two classic projections — Mercator and equirectangular (plate carrée) — with the WGS84 semi-major axis R = 6,378,137 m.
Enter a point, choose a central meridian λ₀ (where x = 0), and for equirectangular a standard parallel φ₁. The tool is forward-only and uses spherical formulas, which makes it ideal for learning how projections behave — not for survey-grade GIS work, where ellipsoidal series are used instead.
Formula: Spherical Mercator: x = R·(λ − λ₀), y = R·ln(tan(π/4 + φ/2)), R = 6,378,137 m (WGS84 semi-major axis). Forward projection only. Spherical formulas, intended for education — real GIS software uses ellipsoidal series (e.g. the formulas in the WGS84 definition).
How to use this tool
Type the latitude and longitude in decimal degrees. The Paris example (48.8584, 2.2945) is pre-filled so you can see a result immediately.
Choose Mercator or equirectangular, set the central meridian λ₀ (0° for Greenwich-centred maps), and for equirectangular the standard parallel φ₁.
The projected easting (x) and northing (y) appear in metres with kilometre equivalents, alongside the exact formula applied.
What people use this tool for
Students project the same point with both formulas and watch Mercator's y stretch logarithmically while equirectangular keeps it linear in latitude.
Plugging in coordinates shows why online maps cut off at ±85.05° latitude — Mercator's y heads to infinity at the poles.
Before touching GIS software, learners see what “forward projection” means concretely: angles in, metres out.
How it's calculated
Spherical forward formulas with R = 6,378,137 m (WGS84 semi-major axis). Mercator: x = R·(λ − λ₀), y = R·ln(tan(π/4 + φ/2)). Equirectangular: x = R·(λ − λ₀)·cos φ₁, y = R·φ; all angles in radians. Forward-only (no inverse). Latitudes beyond ±85.05112878° are rejected for Mercator since y → ∞ at the poles. Educational accuracy: matches ellipsoidal web-map formulas to within metres at city scale.
Related tools and resources
Frequently asked questions
Why does Mercator refuse latitudes beyond ±85.05°?
Because y = R·ln(tan(π/4 + φ/2)) grows without bound as latitude approaches ±90° — the poles would project to infinity. ±85.0511° is the Web Mercator cutoff that all online slippy maps use, so this tool matches them.
Why is x identical in both projections when the standard parallel is 0°?
At φ₁ = 0°, cos φ₁ = 1, so equirectangular's x = R·(λ − λ₀)·1 is exactly Mercator's x. The two projections only differ in y: logarithmic stretching in Mercator versus linear-in-latitude in equirectangular.
Are these the formulas real web maps use?
Close but not identical: production web maps use ellipsoidal Mercator series on the WGS84 ellipsoid for sub-metre precision. The spherical forms here agree with them to within a few metres at city scale, which is why textbooks teach these first.
What is the central meridian λ₀ for?
It sets the longitude where x = 0, i.e. where the map is centred. Keep 0° for a Greenwich-centred world map; a regional map might use its own central longitude so that x values stay small and readable.
Data sources & methodology
- Wikipedia — Mercator projection — Spherical Mercator formulas x = R(λ−λ₀), y = R·ln(tan(π/4+φ/2)).
- Wikipedia — Equirectangular projection — Formulas x = R(λ−λ₀)cos φ₁, y = Rφ.
- Wikipedia — Map projection — Background on forward projection and the distortions involved.