Geography Reference
In-Depth Information
To Cartesian coordinates ( x, y ) R
2 of the chart { R
2 ,δ} of the ellipsoid of revolution E
A 1 ,A 2 ,
A 1 R
is equipped the metric topology of a two-dimensional
Euclidean space (“flat space”). The functions {x ( L − B ) ,y ( L − B ) } areassumedtobeanalytical.
If the distortion energy
+ , R
+
A 2 <A 1 .Thechartof E
A 1 ,A 2
dS l tr C l G l = min
1
2
(22.2)
with respect to
West-East longitude L W , L E and South-North Latitude B S , B N of type fixed boundaries
In particular over a Reference Meridian Strip
{
L W
L
L E ,B S
B
B N }
E 2 (sin B ) 2 2 cos B x L
= in
x ( L,B ) ,y ( L,B )
L E
dL B N
B S
E 2 )
G 11 + x 2 B
G 22 + y L
G 33 + y B
1
2
A 1 (1
dB
(22.3)
1
G 44
L W
L E
dL B N
B S
E 2 (sin B ) 2 2 cos B Grad x
= in
x ( L,B ) ,y ( L,B )
1
2
A 1 (1 − E 2 )
2 + Grad y
2
dB
(22.4)
1
L W
Is assumed to be minimal, the mapping equations, namely the functions x ( L, B ) ,y ( L, B )are
harmonic,
Δ ( L, B )=0 ,Δy ( L, B ) = 0
(22.5)
1
G 11
+
1
G 22
∂L
∂L
∂B
∂B
Δ :=
(22.6)
Is the Laplace-Beltrami operator in Gauss ellipsoidal coordinates subject to
G l = N 2 cos 2 B 0
M 2
(22.7)
0
C l = x L + y L
x L x B + y L y B
(22.8)
x 2 B + y B
x L x B + y L y B
The coordinates of the left metric tensor G l and the coordinates of the left Cauchy-Green defor-
mation tensor C l ,where
E 2 )(1
E 2 sin 2 B ) 3 / 2
M := A 1 (1
(22.9)
N := A 1 (1 − E 2 sin 2 B ) 1 / 2
(22.10)
are the principal radii, also called “ meridianal ”and“ normal ”. E 2 := ( A 1 − A 2 ) /A 1 denotes the
relative eccentricity of E
A 1 ,A 2 .Its left surface element
dS l := dLdB det G l = dL dB
A 1 (1 − E 2 )
1 − E 2 sin 2 B 2
(22.11)
has been represented un terms of Gauss ellipsoidal coordinates ( L , B ).
End of Theorem.
Though details for the proof of Theorem 22.1 are given in Appendix 1, we like to give here
a short review of the metric topology of the left manifold
2
M
r ,namelythe ellipsoid-of-revolution
2
A 1 ,A 2
2
2 ,inTable 22.2 .Bymeans
E
,inTable 22.1 and of the right manifold
M
r ,namelythe plane
P
2
A 1 ,A 2
of Eq.( 22.12 ) we define the algebraic geometry of the ellipsoid
E
of rotational symmetry. The
 
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