Geography Reference
In-Depth Information
G
Generalized Mollweide Projection
Generalized Mollweide projection of the ellipsoid-of-revolution, the standard Mollweide pro-
jection, the generalized Mollweide projection, general equiareal pseudo-cylindrical mapping
equations.
2
The standard Mollweide projection of the sphere
S
R which is of type equiareal pseudo-cylindrical
2
A 1 ,A 2
is generalized to the biaxial ellipsoid
E
.
Within the class of pseudo-cylindrical mapping equa-
tions ( G.8 )of
2
A 1 ,A 2
(semi-major axis A 1 , semi-minor axis
A 2 ) it is shown by solving the general eigenvalue problem
(Tissot analysis) that only equiareal mappings, no confor-
mal mappings exist. The mapping equations ( G.20 ), which
generalize those from S
E
2 R to E
A 1 ,A 2
, lead under the equiareal
postulate to a generalized Kepler equation ( G.39 ), which
is solved by Newton iteration, for instance, see Table G.1 .
Two variants of the ellipsoidal Mollweide projection, in
particular, ( G.35 )and( G.36 )versus( G.37 )and( G.38 ),
are presented, which guarantee that parallel circles (coor-
dinate lines of constant ellipsoidal latitude) are mapped
onto straight lines in the plane, while meridians (coordi-
nate lines of constant ellipsoidal longitude) are mapped
onto ellipses of variable axes. A theorem collects the basic
results. Computer graphical examples illustrate the first
pseudo-cylindrical map projection of
2
A 1 ,A 2
E
of type gener-
alized Mollweide .
With advent of artificial satellites of the Earth measuring precisely its size and shape, the ellip-
soidal reference figure becomes more and more obvious. In order to present an equiareal map
of a biaxial reference ellipsoid (which is of central importance for a graphical representation of
environmental data, for example, from remote sensing ) the popular Mollweide projection of the
sphere
2
R of radius R has to be generalized into an equiareal pseudo-cylindrical projection of the
ellipsoid-of-revolution or spheroid
S
2
A 1
,A 2 with the semi-major axis A 1 andwiththesemi-minor
axis A 2 . for example, according to the Geodetic Reference System 1980.
E
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