Geoscience Reference
In-Depth Information
00
12
, in seconds) for different side lengths
(x
2
x
1
) and distances (y
m
) of the side from the central meridian
Table 6.4 Datasheet for the arc-to-chord correction (ʴ
x
2
x
1
(km)
y
m
(km)
0
4
8
12
16
20
24
28
32
36
40
100
0.0
1.0
2.0
3.0
4.0
5.1
6.1
7.1
8.1
9.1
10.1
200
0.0
2.0
4.1
6.1
8.1
10.1
12.2
14.2
16.2
18.3
20.3
300
0.0
3.0
6.1
9.1
12.2
15.2
18.2
21.3
24.3
27.4
30.4
applied to the observed direction. Note that the sign of
should be taken into
account. For instance, in Fig.
6.12b
, as the values of observed directions increase in
the cl
ockwise direction, when converting the geodesic P
1
P
2
to its equivalent chord
P
0
1
P
0
2
, the sign of t
he arc
-to-chord correction
ʴ
ʴ
12
is negative. Similarly, when
converting P
2
P
1
to P
0
2
P
0
1
, the sign of the arc-to-chord correction
ʴ
12
becomes
positive. Represented by seconds, the approximation formula for the arc-to-chord
correction is:
9
=
1:2
ﾼ
ρ
00
y
m
00
ʴ
ð
x
2
x
1
Þ
2R
m
:
ð
6
:
56
Þ
ﾼ
ρ
00
y
m
2R
m
;
00
2
ʴ
ð
x
2
x
1
Þ
:
1
This formula has an accuracy of better than 0.1
00
and is typically applied to
computations of the third-order triangulation or lower.
It can be seen from (
6.56
) that the value of an arc-to-chord correction is likely to
become larger the further away the side is from the central meridian. Some numeric
values of the arc-to-chord correction calculated according to (
6.56
) are listed in
Table
6.4
.
In Table
6.4
,(x
2
x
1
) roughly corresponds to the side length of the geodetic
network and y
m
is approximately the distance of the side from the central meridian.
It becomes obvious that the arc-to-chord corrections are not negligible for tri-
angulations of various orders.
Precise Formula for Arc-to-Chord Correction
In much of the literature, the approximate formulae are used as differential equa-
tions to derive the relatively precise formulae. The new coordinate systems and
formulae for radii of curvature are introduced to establish the second-order differ-
ential equation, and then the solution is found by solving the differential equation.
The derivation is rather long-winded and therefore will not be discussed in
detail here.
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