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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