Although equal-area, mathematically very simple, and preserving parallel spacing, the previous sinusoidal/Sanson-Flamsteed projection is not completely satisfactory at high latitudes, due to excessive shearing and crowded meridians. Craster's parabolic projection has meridians a bit more rounded, but the poles are still sharp. A slightly more complicated analysis leads to Mollweide's projection.
|Plan of Mollweide's projection|
Suppose the equatorial aspect of an equal-area projection with the following properties:
Since the projection is pseudocylindrical with predetermined meridian shapes, let us repeat the approach for determining the equations of the parabolic design: for any parallel, find an ordinate that equates corresponding areas on map and Earth.
Consider an ellipse centered on the origin, with major axis on the -axis:
The area between the -axis and the parallel mapped into is
For , let : ,
for some .
|The area bounded by the Equator and another parallel|
Because , the area of the full ellipse is
The area of a spherical Earth is , therefore
From the development of the sinusoidal projection, we know that the region on a sphere bounded by the Equator and a parallel is a spherical zone with area
Unfortunately, unlike for Craster's, there is not a closed algebraic solution that directly converts (via ) to . We must resort to numerical root solving, which essentially comprises repeatedly "guessing" approximate values for and evaluating differences until a desired precision is achieved. This task is ideally suited to electronic computers; previously, human "computers" (the original meaning of the word) composed interpolation tables by laboriously calculating values for selected latitudes. Nevertheless, iterative numerical algorithms like the secant and Newton-Raphson methods converge relatively quickly if the initial guess is about itself, except near - but not at - the poles.
Finally, from the ellipse equation, the Eastern boundary meridian is given by
Like in all pseudocylindrical designs, , therefore the equations for Mollweide's projection are: