422 20 Geodetic mapping
Corollary 20.1 (κ g , κ n , τ g ).
κ
2
g = G M 1 M 2 [U
M 1 + U
K 1 U
L 1 {
M 1
K 1 L 1
}][U
M 2 + U
K 2 U
L 2 {
M 2
K 2 L 2
}] ,
(20.28)
κ n = H KL U
K U
L ,
(20.29)
τ g = [H KL U
K U
L + {
N
KL }H NM U
K U
L U
M ]×
×[G M 1 M 2 [U
M 1 + U
K 1 U
L 1 {
M 1
K 1 L 1
}][U
M 2 + U
K 2 U
L 2 {
M 2
K 2 L 2
}]]
−1/2 .
(20.30)
End of Corollary.
The curve C is called geodesic if κ g = 0 and the curve C is called a geodesic circle if κ g = const.,
κ n = const., and τ g = 0. Compare with Examples 20.1 and 20.2.
Example 20.1 (Geodesic as a submanifold in {E
2
A 1 ,A 2
, G KL }, L = c = const: “meridian”).
X = +E 1
A 1 cos T cos c
1 − E 2 sin
2 T
+ E 2
A 1 cos T sin c
1 − E 2 sin
2 T
+ E 3
A 1 (1 − E
2 ) sin T
1 − E 2 sin
2 T
,
(20.31)
D 1 = −E 1 sin T cos c − E 2 sin T sin c + E 3 cos T =: ˙
X/ ˙
X ,
D 2 = E 1 sin c − E 2 cos c ,
D 3 = E 1 cos T cos c + E 2 cos T sin c + E 3 sin T ,
(20.32)
D =
⎡
⎢
⎢
⎣
− sin T cos c − sin T sin c cos T
sin c
− cos c
0
cos T cos c
cos T sin c sin T
⎤
⎥
⎥
⎦ E ,
(20.33)
D = RE , D
= R
E = R
R
T D = ΩD ∀ R ∈ SO(3) ,
(20.34)
R
=
⎡
⎢
⎢
⎣
−T
cos T cos c −T
cos T sin c −T
sin T
0
0
0
−T
sin T cos c −T
sin T sin c +T
cos T
⎤
⎥
⎥
⎦ , Ω := R
R
T =
⎡
⎢
⎢
⎣
0 0 −T
0 0 0
+T
0 0
⎤
⎥
⎥
⎦ ,
(20.35)
κ g = 0 , κ n = −T
, τ g = 0 ,
˙
X =
dS
dT
=
A 1 (1 − E
2 )
(1 − E 2 sin
2 T ) 3/2 , T
:=
dT
dS
=
(1 − E
2 sin
2 T )
3/2
A 1 (1 − E 2 )
,
(20.36)
κ g = 0 , κ n = −
(1 − E
2 sin
2 T )
3/2
A 1 (1 − E 2 )
, τ g = 0 .
(20.37)
End of Example.
Obviously, the meridian L = const. is a geodesic.