Geography Reference
In-Depth Information
Example 20.2 (Geodesic as a submanifold in { E
A 1 ,A 2 ,G KL },B = c = const: “parallel circle”).
X =+ E 1 A 1 cos c cos T
+ E 2 A 1 cos c sin T
+ E 3 A 1 (1 E 2 )sin c
1
1
1
,
(20.38)
E 2 sin 2 c
E 2 sin 2 c
E 2 sin 2 c
X /
X
D 1 =
E 1 sin T + E 2 cos T =:
,
D 2 =
E 1 sin c cos T
E 2 sin c sin T + E 3 cos c,
(20.39)
D 3 =+ E 1 cos c cos T + E 2 cos c sin T + E 3 sin c,
sin T cos T 0
sin c cos T − sin c sin T cos c
cos c cos T
E ,
D =
cos c sin T sin c
D =R E , D =R E =R R T D D R SO(3) ,
T cos T
T sin T
0
,
R =
+ T sin c sin T
T sin c cos T 0
(20.40)
T cos c sin T + T sin c cos T 0
0 T sin c
T cos c
,
Ω:=R R T =
T sin c
0
0
+ T cos c
0
0
κ g =+ T sin c, κ n = −T cos c, τ g =0 ,
= dS
A 1 cos c
X
1 E 2 sin 2 c
dT =
,
d S = 1
E 2 sin 2 c
A 1 cos c
T := d T
(20.41)
κ g =+ 1
E 2 sin 2 c
A 1
tan c =const ,
1 E 2 sin 2 c
A 1
κ n =
=const ,
τ g =0 .
End of Example.
2
R great circles
are geodesics , but small circles are geodesic circles . Following the curvature measure representation
in Corollary 20.1 , we can characterize geodesics and geodesic circles by differential equations.
Obviously, the parallel circle B =const.isa geodesic circle . Note that for a sphere
S
 
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