Geography Reference
In-Depth Information
There are two variants of interest in order to determine the axes
a
and
b
of the ellipse that is defined
by
x
2
/a
2
(
Λ
)+
y
2
/b
2
= 0. (i) Variant one is being motived by the original Mollweide projection for
the sphere
2
R
. Accordingly, we define (
G.35
)and(
G.36
). If we set
E
=0,the
spherical Mollweide
projection
is derived. (ii) An alternative variant is the postulate of an equidistant mapping of the
equator of
S
2
A
1
,A
2
E
according to (
G.37
)and(
G.38
).
b
:=
A
1
√
2
(G.35)
ln
1+
E
1
E
2
;
E
2
)
a
=
A
1
(1
−
2
E
πE
√
2
(
G.34
), (
G.35
)
⇔
E
+
(G.36)
−
1
−
x
(
Φ
=0)=2
πa
:= 2
πA
1
,y
(
Φ
=0)=0
⇔
(G.37)
a
:=
A
1
,
(
G.34
), (
G.37
)
⇔ b
=
A
1
(1
−
E
2
)
πE
ln
1+
E
1
E
2
.
2
E
E
+
(G.38)
−
1
−
As soon as we implement the axe
a
and the axe
b
in the
generalized Kepler equation
(
G.31
), we gain
its final form (
G.39
). We here note the symmetry of the right-side representation. Table
G.1
is a
Newton iteration solution
(see
Toernig 1979
)), for instance) of the generalized Kepler equation for
the biaxial ellipsoid
A
1
,A
2
for the axes
A
1
and
A
2
as well as the relative eccentricity
E
according
to the Geodetic Reference System 1980 (Bulletin Geodesique 58 (1984) pp. 388-398).
E
2
t
+ sin 2
t
=
π
ln
1+
E
sin
Φ
2
E
sin
Φ
1
−E
sin
Φ
+
E
2
sin
2
Φ
1
−
.
(G.39)
ln
1+
E
2
E
1
−E
2
1
−E
+
Tab l e G. 1
Newton iterative solution of the generalized Kepler equation. Parameters:
A
1
=6
,
378
,
137 m
,A
2
=
6
,
356
,
752
.
3141 m
,E
2
=0
.
006694380002290
t
(
◦
)
Φ
t
(rad)
90
◦
1.56673055580
89.767053190
80
◦
1.23781233200
70.921424475
70
◦
1.03751932140
59.445479975
60
◦
0.86517291851
49.570758193
50
◦
0.70717579230
40.518189428
40
◦
0.55790136480
31.965394499
30
◦
0.41428104971
23.736556358
20
◦
0.27434065788
15.718562294
10
◦
0.13663879358
7.828826413
0
◦
0.00000000000
0.000000000
Before we present examples of the ellipsoidal Mollweide projection, we briefly summarize the
basic results in Theorem
G.4
.
Theorem G.4 (Generalized Mollweide projection of the biaxial ellipsoid
E
A
1
,A
2
).
A
1
,A
2
In the class of pseudo-cylindrical mappings of the biaxial ellipsoid
E
,Eq.(
G.40
) generate an
equiareal mapping.
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