Geography Reference
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introducing “surface normal” ellipsoidal longitude
L
as well as “surface normal” ellipsoidal latitude
B
,where
E
2
:= (
A
1
−
A
2
)
/A
1
=1
A
2
/A
1
denotes the first relative eccentricity. According to the
−
relation
L, B
∈{
0
≤
L
≤
2
π,
−
π/
2
<B<π/
2
}
, we exclude from the domain
{
L, B
}
North and
. A minimal atlas of E
A
1
,A
2
based
on two charts and which covers all points of the ellipsoid-of-revolution is given by E. Grafarend
and R. Syffus (1995), in great detail.
Conformal coordinates
x
and
y
(isometric coordinates, isothermal coordinates) are constructed
from the “surface normal” ellipsoidal coordinates
L
and
B
as solutions of the Korn-Lichtenstein
equations (conformal change from one chart to another chart, c:cha-cha-cha)
A
1
,A
2
South Pole. Thus,
{
L, B
}
constitutes only a first chart of
E
x
L
x
B
=
−
y
L
y
B
,
1
G
11
G
22
−
G
12
G
11
(16.6)
−
G
22
G
12
G
12
subject to the integrability conditions
x
LB
=
x
BL
and
y
LB
=
y
BL
or
LB
x
:=
G
11
x
B
−
G
12
x
L
+
G
22
x
L
−
G
12
x
B
G
11
G
22
G
11
G
22
=0
,
G
12
G
12
−
−
B
L
(16.7)
LB
y
:=
G
11
y
B
+
G
22
y
L
−
G
12
y
L
−
G
12
y
B
G
11
G
22
−
G
11
G
22
−
=0
,
G
12
G
12
B
L
and
x
L
y
L
x
B
y
B
>
0
(16.8)
(orientation preserving conformeomorphism),
:=
G
11
G
12
{
g
μν
}
∀
μ, ν
∈{
1
,
2
}
G
12
G
22
(16.9)
2
(metric of the first fundamental form of
E
A
1
,A
2
)
.
LB
x
=0and
LB
y
= 0, respectively, are called the
vectorial Laplace-Beltrami equations
.We
here note that a Jacobi map (
16.6
) can be made unique by a proper boundary condition, e.g. the
equidistant map of a particular coordinate line. Examples are equidistant mappings of the circular
equator (Mercator projection) or of the elliptic meridian (transverse Mercator projection). Fur-
thermore, we here note that only few solutions of the Korn-Lichtenstein equations (
16.6
) subject
to the integrability condition (
16.7
) (vectorial Laplace-Beltrami equations) and the condition of
orientation preservation are known. We list two in the following.
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