G Generalized Mollweide projection
Generalized Mollweide projection of the ellipsoid-of-revolution, the standard Mollweide projection, the
generalized Mollweide projection, general equiareal pseudo-cylindrical mapping equations.
The standard Mollweide projection of the sphere S
2
R which is of type equiareal pseudo-cylindrical is
generalized to the biaxial ellipsoid E
2
A 1 ,A 2
.
Important!
Within the class of pseudo-cylindrical mapping equations (G.8) of E
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 conformal mappings exist. The mapping equations
(G.20), which generalize those from S
2
R to E
2
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 (coordinate lines of constant ellipsoidal latitude) are mapped onto straight lines in the plane,
while meridians (coordinate 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 E
2
A 1 ,A 2
of type generalized Mollweide.
With advent of artificial satellites of the Earth measuring precisely its size and shape, the ellipsoidal
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 S
2
R of radius
R has to be generalized into an equiareal pseudo-cylindrical projection of the ellipsoid-of-revolution or
spheroid E
2
A 1 ,A 2
with the semi-major axis A 1 and with the semi-minor axis A 2 , for example, according
to the Geodetic Reference System 1980.
Section G-1.
In order to generalize the standard Mollweide projection of S
2
R towards E
2
A 1 ,A 2
(compare with
J. P. Snyder (1977, 1979)), Section G-1 is a setup of general pseudo-cylindrical mapping equations of
class (G.8) which allow only equiareal, but no conformal projections. (We met the same situation for
S
2
R .) By solving a general eigenvalue–eigenvector problem, the principal distortions as well as their directions (eigenvectors) are computed, in particular, on the basis of the metric of the plane represented
by pullback in terms of the left Cauchy–Green deformation tensor and of the metric of the spheroid.
Section G-2.
In Section G-2, we specialize the ellipsoidal Mollweide projection by an “Ansatz” (G.20), leading to
the problem to solve a generalized Kepler equation (G.39), for example, by Newton iteration, see
Table G.1. The basic results are collected in three corollaries and one theorem.
Section G-3.
Finally, completing the preceding Sections G-1 and G-2, Section G-3 presents computer graphics of
the generalized Mollweide projection of E
2
A 1 ,A 2
.
G-1 The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane
First, we construct a minimal atlas of the biaxial ellipsoid E
2
A 1 ,A 2
(“ellipsoid-of-revolution”, “spheroid”)
based on local coordinates of type {longitude, latitude} and {meta-longitude, meta-latitude}.
Generalized Mollweide projection of the ellipsoid-of-revolution, the standard Mollweide projection, the
generalized Mollweide projection, general equiareal pseudo-cylindrical mapping equations.
The standard Mollweide projection of the sphere S
2
R which is of type equiareal pseudo-cylindrical is
generalized to the biaxial ellipsoid E
2
A 1 ,A 2
.
Important!
Within the class of pseudo-cylindrical mapping equations (G.8) of E
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 conformal mappings exist. The mapping equations
(G.20), which generalize those from S
2
R to E
2
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 (coordinate lines of constant ellipsoidal latitude) are mapped onto straight lines in the plane,
while meridians (coordinate 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 E
2
A 1 ,A 2
of type generalized Mollweide.
With advent of artificial satellites of the Earth measuring precisely its size and shape, the ellipsoidal
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 S
2
R of radius
R has to be generalized into an equiareal pseudo-cylindrical projection of the ellipsoid-of-revolution or
spheroid E
2
A 1 ,A 2
with the semi-major axis A 1 and with the semi-minor axis A 2 , for example, according
to the Geodetic Reference System 1980.
Section G-1.
In order to generalize the standard Mollweide projection of S
2
R towards E
2
A 1 ,A 2
(compare with
J. P. Snyder (1977, 1979)), Section G-1 is a setup of general pseudo-cylindrical mapping equations of
class (G.8) which allow only equiareal, but no conformal projections. (We met the same situation for
S
2
R .) By solving a general eigenvalue–eigenvector problem, the principal distortions as well as their directions (eigenvectors) are computed, in particular, on the basis of the metric of the plane represented
by pullback in terms of the left Cauchy–Green deformation tensor and of the metric of the spheroid.
Section G-2.
In Section G-2, we specialize the ellipsoidal Mollweide projection by an “Ansatz” (G.20), leading to
the problem to solve a generalized Kepler equation (G.39), for example, by Newton iteration, see
Table G.1. The basic results are collected in three corollaries and one theorem.
Section G-3.
Finally, completing the preceding Sections G-1 and G-2, Section G-3 presents computer graphics of
the generalized Mollweide projection of E
2
A 1 ,A 2
.
G-1 The pseudo-cylindrical mapping of the biaxial ellipsoid onto the plane
First, we construct a minimal atlas of the biaxial ellipsoid E
2
A 1 ,A 2
(“ellipsoid-of-revolution”, “spheroid”)
based on local coordinates of type {longitude, latitude} and {meta-longitude, meta-latitude}.