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the geodesic distance S. The features of this type of formulae are that the
accuracy of solution is distance-dependent, and the longer the distance the
slower the rates of convergence. Converge may not even occur, so finding
solutions would be impossible. Hence, this approach is better applicable to
short distances.
2. We take an auxiliary sphere, convert the ellipsoidal elements to the spherical
surface, solve the problem by applying the formula for spherical triangles on a
spherical surface, and then reduce the computed results back to the ellipsoid.
Because the difference between the ellipsoid and spheroid is only a very small
flattening, the expressions for transformation of the corresponding elements
between the ellipsoidal surface and the spherical surface only involve the
ascending power series of the small quantities e
2
(eccentricity squared) or e
02
.
The Bessel's formula studied in this section can represent this type of formula.
Such formulae are independent of distance and can be applied to the solution of
geodetic problems over any distances.
3. Take advantage of numerical integration and solve the direct and inverse
geodetic problems on the ellipsoid. In general, the solutions are composed of a
strict solution for the sphere plus a correction to the ellipsoid, determined by
numerical integration. By employing numerical integration, routines that are
usually available in current computer software like MATLAB, the problems of
classical geodesy are easily solved to the desired accuracy (Sj¨berg et al. 2012).
The accuracy of solutions of geodetic problems depends on different practical
matters. Take the adjustment of the astro-geodetic network for instance. The
relative mean square error of the side lengths in the first-order triangulation chain
m
S
1. 0
00
, i.e.,
m
A
S
ﾼ
1
:
200000 and the accuracy of azimuth m
A
ﾼ
ˁ
1
:
200000
:
We set S
20 km and let the coordinate components be the same as the longitu-
dinal and lateral errors, namely:
ﾼ
m
x
ﾼ
m
y
ﾼ
0
:
1m
:
After considering the adjustment of the first-order triangulation chain, the
accuracy of the point positioning can be slightly improved. Putting m
x
ﾼ
m
y
45
, we obtain:
ﾼ
0.09 m, if expressed by geodetic coordinates, when B
ﾼ
m
x
M
ˁ
00
003
00
,
m
B
ﾼ
ﾼ
0
:
m
y
N cos B
ˁ
00
004
00
m
L
ﾼ
ﾼ
0
:
:
When deciding accuracy requirements for the solution formulae, one should
generally adhere to the following principle: it has to be ensured that the computa-
tional errors introduced by the formulae do not have any further effect on the real
accuracy of the field observations and adjusted values. In accordance with this
principle, considering the possible accumulation of errors in the pointwise
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