Geoscience Reference
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This is Simpson's rule for the approximate integration.
According to Simpson's rule for integration, we divide the interval of integration
in (
6.67
) into two subintervals of equal width
2
; then:
S
6
m
1
þ
s
ᄐ
ð
4m
m
þ
m
2
Þ
,
ð
6
:
68
Þ
where m
1
and m
2
are the scale factors at points P
1
and P
2
, respectively. m
m
is the
scale factor at the midpoint of the geodesic.
Now we will derive the relationship between s and D. As shown in Fig.
6.16
,
P
1
0
P
2
0
is the projected curve of geodesic P
1
P
2
, approximating a circular arc, where
O is the center of the circular arc, F is the midpoint, and
ʴ
is the curvature
P
1
0
OF is also
correction. Hence
∠
ʴ
. The relationship is given by:
D
2
R
ᄐ
=
D
2R
, R
s
2
sin
ʴ ᄐ
ᄐ
ʴ
:
It follows from the above two equations that:
s sin
ʴ
D
ᄐ
,
ʴ
3
5
5
7
7
ʴ ᄐ ʴ
ʴ
!
þ
ʴ
!
ʴ
and sin
!
þ
,
3
2
40 km, and
ʴ
15 mm, the required side
lengths of the first-order triangulation are defined to maintain an accuracy to
millimeter level, so this term and higher-order terms can all be neglected. As a
result, in computations, one can consider:
with
ʴ ᄐ
30 "
ᄐ
0.00015 rad, s
ᄐ
s
ᄐ
0
:
3
!
D
ᄐ
s
:
ð
6
:
69
Þ
We replace s in (
6.68
) with D to get:
S
6
m
1
þ
D
ᄐ
ð
4m
m
þ
m
2
Þ:
ð
6
:
70
Þ
Again, according to (
6.66
), we have:
y
1
2
y
1
4
24R
1
4
m
1
ᄐ
1
þ
2R
1
2
þ
y
m
2
2R
m
2
þ
y
m
4
24R
m
4
m
m
ᄐ
1
þ
y
2
2
y
2
4
m
2
ᄐ
1
þ
2R
2
2
þ
24R
2
4
:
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