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
=Ω
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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