Geoscience Reference
In-Depth Information
L
ﾼ
LS
ðÞ
, B
ﾼ
BS
ðÞ
, A
ﾼ
AS
ðÞ:
Obviously, the above function can be differentiated repeatedly. Expanding the
above function at point P
1
based on Maclaurin series results in:
0
1
0
1
0
1
d
2
L
dS
2
S
2
2
þ
d
3
L
dS
3
S
3
6
þ
dL
dS
@
A
@
A
@
A
l
ﾼ
L
2
L
1
ﾼ
S
þ
,
0
@
1
A
1
0
@
1
A
1
0
@
1
A
1
1
1
1
d
2
B
dS
2
S
2
2
þ
d
3
B
dS
3
S
3
6
þ
dB
dS
b
ﾼ
B
2
B
1
ﾼ
S
þ
,
ð
5
:
70
Þ
0
1
0
1
0
1
d
2
A
dS
2
S
2
2
þ
d
3
A
dS
3
S
3
6
þ:
dA
dS
@
A
@
A
@
A
180
∘
a
ﾼ
A
2
A
1
ﾼ
S
þ
1
1
1
where the subscript “1” denotes the value when the derivatives of various orders
take S
A
1
). It can thus be seen that
finding the derivatives of different orders in the equations is the key to obtain the
expansions of l, b, and a. Here, three first-order derivatives will form a differential
equation of the geodesic. With N
ﾼ
0 (i.e., the value at point P
1
; B
ﾼ
B
1
, A
ﾼ
c
V
, M
c
V
3
, one obtains:
ﾼ
ﾼ
dL
dS
ﾼ
V
c
sec B sin A,
V
3
c
dB
dS
ﾼ
cos A,
dA
dS
ﾼ
V
c
tan B sin A
:
Taking repeated derivatives of the above equations results in the derivatives of
various orders in (
5.70
); hence, one will get the power series in geodesic distance
S expanded from the differences in longitude l, latitude b, and azimuth a, generally
known as the Legendre series.
Legendre series converge more slowly. However, if l, b, and a are expanded into
the power series in S at the midpoint of the geodesic P
S/2
instead of the end point of
the geodesic P
1
, then the convergence rate of the series will increase considerably.
Expanding the difference in geodetic longitude at midpoint P
S/2
based on the Taylor
series yields:
Search WWH ::
Custom Search