Geography Reference
In-Depth Information
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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