Cartographic Projection
every map is wrong, and by exactly how much
Figure

What reprojecting from Mollweide to Mercator does, drawn on the Mollweide plane

Drawn here at the parameters it defaults to, with every essay that calls it.
What reprojecting from Mollweide to Mercator does, drawn on the Mollweide plane. Every dot is a point of the source picture, sized by the angular deformation the reprojection itself applies there — not the deformation of either projection against the sphere, but of the map that carries one plane to the other. It reaches 116.5° here. This is the quantity a reprojection actually costs a shape, and it is computed from the two Jacobians alone: the sphere's own metric cancels exactly, so the transformation does not depend on the shape of the Earth at all. Drawn on the source plane, in Mollweide.

Every dot is a point of the source picture, sized by the angular deformation the reprojection itself applies there — not the deformation of either projection against the sphere, but of the map that carries one plane to the other. It reaches 116.5° here. This is the quantity a reprojection actually costs a shape, and it is computed from the two Jacobians alone: the sphere's own metric cancels exactly, so the transformation does not depend on the shape of the Earth at all. Drawn on the source plane, in Mollweide.

It is one of 14 figures drawn by the same construction, and the drawing above is this one with nothing changed.

1 essay calls it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The circle a reprojection is about to turn into an ellipse. A circle drawn on the Mollweide plane, and its image under the map that carries that plane to Mercator — normalised so each pair has the same area, because the whole map can be rescaled and the shape is what is being shown. The dashed circle is what an undistorted reprojection would leave. This is Tissot's construction with a plane in place of the sphere, and it is the right picture for a reprojection because both ends are pictures. Worst angular deformation over the sampled points: 81.2°. Drawn on the source plane, in Mollweide. Measuring distortion

A projection between two projections

Every distortion measured on this site so far compares a map with the sphere. The operation a machine actually performs compares a map with another map — and that map has its own two principal scales, its own areal factor and its own angular deformation, none of which is the difference of the two it was built from.

The whole library