Map Projections

### Deducing Mollweide's Projection

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:

• a world map is bounded by an ellipse twice as broad as tall
• parallels of latitude map into parallel straight lines with uniform scale
• the central meridian is a straight standard line; all other ones are semielliptical arcs, symmetrical around the Equator and the central meridian

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:

For .
The area between the -axis and the parallel mapped into is

For , let : ,

Since

and

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

Making ,

 Mollweide's projection

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:

 www.progonos.com/furuti    January 28, 2013