Geography Reference
In-Depth Information
Section G-1.
2
R
towards
A
1
,A
2
(compare with
Sny-
der 1977
,
1979a
), Sect.
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
In order to generalize the standard Mollweide projection of
S
E
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 rep-
resented by
pullback
in terms of the left Cauchy-Green deformation tensor and of the metric of
the spheroid.
S
Section G-2.
In Sect.
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 Sects.
G-1
and
G-2
, Sect.
G-3
presents computer graphics of
the generalized Mollweide projection
2
A
1
,A
2
E
.
G-1 The Pseudo-Cylindrical Mapping of the Biaxial Ellipsoid Onto
the Plane
2
A
1
,A
2
First, we construct a
minimal atlas
of the
biaxial ellipsoid
E
(“ellipsoid-of-revolution”,
“spheroid”) based on local coordinates of type
{
longitude, latitude
}
and
{
meta-longitude, meta-
latitude
}
.
2
A
1
,A
2
X
∈
E
:=
3
:=
X
+
,A
1
>A
2
,
X
2
+
Y
2
A
1
+
Z
2
+
,A
2
∈
R
A
2
=1
,A
1
∈
R
∈
R
(G.1)
X
(
L, B
)=
E
2
)sin
Φ
=
e
1
A
1
cos
Φ
cos
Λ
+
e
2
A
1
cos
Φ
sin
Λ
+
e
3
A
1
(1
−
1
1
1
.
(G.2)
E
2
sin
2
Φ
E
2
sin
2
Φ
E
2
sin
2
Φ
−
−
−
For surface normal ellipsoidal
{
longitude
Λ
, latitude
Φ
}
,wechoosetheopenset0
<Λ<
2
π
and
2
A
1
,A
2
−
π/
2
<Φ<
+
π/
2 in order to avoid any
coordinate singularity
once we endow the manifold
E
2
A
1
,A
2
with a differentiable structure. Indeed, (
G.2
) covers all points of
E
except the meridians
Λ
=0
and
Λ
=
π
, respectively, as well as the poles
Φ
=
±
π/
2
.E
denotes the relative eccentricity of the
2
A
1
,A
2
defined by
E
2
:= (
A
1
−
A
2
)
/A
1
. In order to guarantee
bijectivity
of the
biaxial ellipsoid
E
mapping
{
X,Y,Z
} →{
Λ, Φ
}
,weuse
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