Geography Reference
In-Depth Information
Box 20.3 (The Legendre recurrence of
U
(
m
)
A
in terms of
U
A
, index set
A
1
,A
2
,...,A
m−
1
,A
m
∈{
1
,
2
}
(
Legendre 1806
)).
A
A
1
A
2
U
A
1
U
A
2
(geodesic
,
(
20.42
))
,
U
A
=
−
(20.55)
U
A
=
A
A
1
A
2
U
A
3
U
A
2
U
A
1
+2
A
A
A
3
A
4
U
A
2
U
A
3
U
A
4
,
=
−
A
1
A
2
A
3
U
(4)
A
=
(20.56)
A
A
1
A
2
,
A
3
A
4
U
A
4
U
A
3
U
A
2
U
A
1
=
−
+3
A
,
A
3
A
3
A
4
A
5
U
A
1
U
A
2
U
A
4
U
A
5
A
1
A
2
+
A
,
A
3
A
1
A
4
A
5
A
A
2
U
A
3
U
A
4
U
A
5
A
1
A
2
+6
A
A
1
A
3
A
4
A
3
A
5
A
6
U
A
2
U
A
4
U
A
5
U
A
6
,
etc
.
A
1
A
2
If we replace
U
(
m
)
A
in the Taylor expansion (
20.54
) by means of the Legendre recurrence of
Box
20.3
,wehavesolved
the initial value problem
of the geodesic (
20.42
) in terms of power series
U
A
0
,U
A
0
U
A
0
,U
A
0
U
A
0
U
A
0
etc. generating
an exponential map
, in particular
U
A
(
S
) =
(20.57)
=
U
0
+
SU
0
+
S
2
A
A
1
,A
2
U
A
0
U
A
0
+
...
n
S
m
A
A
1
A
2
...A
m
U
A
0
U
A
0
...U
A
0
.
+ lim
n→∞
m
=3
With respect to the
first chart
, part of the minimal atlas of
E
A
1
,A
2
, namely the orthogonal coor-
, we transform
U
0
via
dinates
{
L, B
}
x
y
S
√
G
22
L
0
=
U
1
0
=
S
√
G
11
,B
0
=
U
2
0
=
(20.58)
into (Riemann) normal coordinates
{x, y}
(inverse Riemann mapping, inverse Riemann cha-cha-
cha). Accordingly, we succeed to represent th
e Ta
ylor series (
20.
54
) i
n terms of (Riemann) normal
coordinates
{x, y}
, in particular,
x
1
:=
x/
√
G
11
and
x
2
:=
y/
√
G
22
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