Geography Reference
In-Depth Information
Corollary 20.2 (Geodesics, geodesic circles).
Acurve
C
(
S
)isa
geodesic
if and only if
U
M
+
M
U
K
U
L
=0
.
(20.42)
KL
Acurve
C
(
S
)isa
geodesic circle
if and only if
U
M
+
G
KL
U
K
U
L
U
M
+3
M
U
K
U
L
KL
L
PQ
U
K
U
P
U
Q
U
M
+
+2
G
KL
(20.43)
+
M
,P
+
Q
M
QP
U
K
U
L
U
P
KL
KL
K
PQ
L
ST
U
P
U
Q
U
S
U
T
U
M
=0
.
+
G
KL
End of Corollary.
. which is spanned by the two tangent vectors
G
1
and
G
2
(
G
1
and
G
2
are neither orthogonal nor normalized),
C
1
and
C
2
(orthonormal
Car-
tan frame
), or
D
1
and
D
2
(orthonormal
Darboux frame
)atthepoint
U
0
=
In the
tangent space
{
T
U
E
A
1
,A
2
,G
KL
}
{
U
0
,U
0
}
(e.g.
{
), we define (
Riemann
)
polar coordinates
and
normal coordinates
by (
20.44
),
in particular, referring to (
20.45
) called
L
0
,B
0
}
or
{
U
0
,V
0
}
{
“eastern”/“right”/“horizontal”
}
and (
20.46
) called
{
“northern”/“up”/“vertical”
}
.
x
=
r
cos
α, y
=
r
sin
α,
(20.44)
∂
X
∂L
G
1
=
=:
C
1
,
(20.45)
∂
X
∂L
G
1
∂
X
∂B
G
2
=
=:
C
2
.
(20.46)
∂
X
∂B
G
2
The polar coordinate
α
is called “
East azimuth
” (ninety degrees
minus
“North azimuth” or
minus ninety degrees plus “South azimuth”/“astronomical azimuth”) while
r
characterizes the
Euclidean distance of a point in
2
A
1
,A
2
U
1
,U
2
{
T
U
E
,G
KL
}
with respect to the origin
{
}
. Figure
20.3
2
A
1
,A
2
U
0
,U
0
}
illustrates the tangent space
{
T
U
0
E
,G
KL
}
at the point
U
0
=
{
. Furthermore, Fig.
20.3
illustrates the Cartan two-leg
. In contrast, Table
20.1
summarizes the
various definitions of polar and normal coordinates with respect to alternative azimuth definitions.
{
C
1
(East),
C
2
(North)
}
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