Eckert VI projection calculator

It solves the projection’s pseudocylindrical auxiliary angle, then applies the equal-area forward equations using your chosen sphere radius and central meridian. The result also includes the solved auxiliary angle in degrees, making the calculation easy to inspect, reproduce, or compare with GIS software and cartographic references.

How the Eckert VI forward projection works

Eckert VI is a pseudocylindrical equal-area projection designed for complete world maps. The latitude does not enter the final coordinate equations directly. First, the calculator converts latitude φ to radians and finds an auxiliary angle θ that satisfies θ + sin θ = (1 + π/2) sin φ. That implicit relationship has one solution between −π/2 and π/2 for every valid latitude. This implementation finds it with a bounded bisection procedure, which is deterministic and remains stable at high latitudes. It then computes x = 2Rλ(1 + cos θ)/√(2 + π) and y = 2Rθ/√(2 + π), where R is the sphere radius and λ is longitude relative to the selected central meridian. Because the radius multiplies both axes, a radius of one gives normalized map coordinates, while a radius expressed in meters or kilometers produces coordinates in that same unit. The returned auxiliary angle is useful for checking intermediate work and explaining why parallels become progressively closer toward the poles.

Choosing inputs, units, and precision

Enter latitude and longitude as decimal degrees. Latitude must remain from −90 through 90, and longitude from −180 through 180. The optional central meridian sets the longitude that maps to x = 0; it defaults to Greenwich at zero degrees. For example, choosing a central meridian near the middle of a region can keep that region close to the center of the resulting world map. The radius defaults to one, which is convenient for mathematical comparisons and normalized graphics. Set it to a positive Earth radius if you need linear coordinates, remembering that Eckert VI here assumes a sphere rather than an ellipsoid. The decimals option controls only the published x, y, and θ values: calculation proceeds at JavaScript floating-point precision before the final rounding step. Values from zero to twelve decimal places are accepted. Longitude differences are left relative to the stated central meridian rather than silently wrapped, so the output directly represents the exact longitude and meridian supplied by the caller. Invalid, missing, non-finite, or out-of-range values return a typed input error.

Interpreting and using the projected coordinates

Use the x and y result as a point in an Eckert VI map plane, not as a universal ground-distance coordinate. The projection preserves area on the sphere, which makes it appropriate for thematic world maps where the relative sizes of regions matter, but shapes, angles, directions, and distances are not preserved everywhere. The equator and central meridian are straight, while other meridians curve and the poles appear as lines rather than single points. A point with longitude equal to the central meridian has x = 0, and a point on the equator has y = 0. If radius is one, combine the normalized point with the same projection constants used by your renderer. If radius is in meters, x and y are in meters, but Euclidean separation between two projected points should not be treated automatically as geodesic distance. This calculator performs the forward spherical transformation only; it does not select a datum, convert an ellipsoid, clip map edges, draw graticules, or invert x and y back to geographic coordinates. API usage is charged from $0.002 per request.