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.
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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:
For .
The area between the -axis and the parallel mapped
into
is
For , let
:
,
Since
and
for some .
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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 ,
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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:
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