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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
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
=
D
2R , R
s
2
sin
ʴ ᄐ
ʴ :
It follows from the above two equations that:
s sin
ʴ
D
,
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 "
0.00015 rad, s
s
0
:
3
!
D
s
:
ð
6
:
69
Þ
We replace s in ( 6.68 ) with D to get:
S
6 m 1 þ
D
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
1
þ
2R 1 2 þ
y m 2
2R m 2 þ
y m 4
24R m 4
m m
1
þ
y 2 2
y 2 4
m 2
1
þ
2R 2 2 þ
24R 2 4 :
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