Geography Reference
In-Depth Information
x
(
Λ, Φ
)=
aΛ
cos
t
(
Φ
)
,y
(
Φ
)=
b
sin
t
(
Φ
)
.
(G.40)
For a given ellipsoidal latitude
Φ
, the reduced latitude
t
is a solution of the transcendental equa-
tion (
G.41
), which for relative eccentricity
E
= 0 coincides with the Kepler equation.
2
t
+ sin 2
t
=
π
ln
1+
E
sin
Φ
2
E
sin
Φ
1
−E
2
sin
2
Φ
1
−E
sin
Φ
+
.
(G.41)
ln
1+
E
2
E
1
−E
2
1
−E
+
There are two variants for th
e
gauge of the axes
a
and
b
of the ellipse
x
2
/a
2
(
Λ
)+
y
2
/b
2
=1
,a
(
Λ
)=
aΛ
. in particular,
b
:=
A
1
√
2
,a
=(
G.36
)(variantone)and
a
:=
A
1
,b
=(
G.38
)(varianttwo).
The principal distortions
Λ
1
and
Λ
2
of the generalized Mollweide projection are given by (
G.24
)
inserting d
t/
d
Φ
accordingto(
G.25
)and
t
(
Φ
), solution of (
G.41
), which are the eigenvalues of the
general eigenvalue problem (
c
AB
Λ
2
G
4
B
)
E
BC
= 0. The principal distortions are plotted along
the eigenvectors which constitute the eigenvector matrix
−
E
BC
}
The coordinate lines
Φ
=const.(
t
= const.) called
parallel circles
are mapped onto straight
lines
y
= const. while the coordinate lines
Λ
= const. called
meridians
are mapped onto ellipses
x
2
/a
2
(
Λ
)+
y
2
/b
2
=1
,a
(
Λ
)=
aΛ
, a straight line for
Λ
= 0 (Greenwich meridian), in particular.
For relative eccentricity
E
= 0, the generalized Mollweide projection of the biaxial ellipsoid
{
A
1
,A
2
E
.
2
variant one, coincides with the standard Mollweide projection of the sphere
S
R
.
End of Theorem.
G-3 Examples
The following six examples illustrate by computer graphics the generalized Mollweide projection
of the biaxial ellipsoid (Figs.
G.1
,
G.2
,
G.3
,
G.4
,
G.5
,and
G.6
).
S
2
R
Fig. G.1.
Standard Mollweide projection of the sphere
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