Geoscience Reference
In-Depth Information
p
cos
2
B
dB,
1
e
2
du
cos
2
u
ﾼ
With (
5.75
), one obtains:
p
1
dB
du
ﾼ
V
2
e
2
:
ð
5
:
77
Þ
Underlying Principle of Bessel's Solution of the Geodetic Problem
Bessel's formula for the solution of the geodetic problem is first to create an
auxiliary sphere with its center at the center of the ellipsoid and the radius of any
length (the problem of spherical triangles is independent of the length of radius) and
then to solve according to the three steps below:
1. Project the ellipsoidal elements onto the spherical surface according to certain
conditions
2. Solve the geodetic problem on the spherical surface
3. Convert the spherical elements obtained into the corresponding ellipsoidal
elements based on the relations of projection
The three conditions of projection for Bessel's solution of the geodetic problem
are as follows:
1. The spherical latitude of the point on the spheroid after projection is equal to the
reduced latitude of the corresponding point on the ellipsoid
2. Projection of the geodesic between two points on the ellipsoid onto the auxiliary
sphere is a great circle
3. The numerical value of the geodetic azimuth A
1
remains unchanged after
projection
In Fig.
5.42
, after the auxiliary sphere is created and the projection is done
according to the three conditions above, there is a geodetic polar triangle NP
1
P
2
on
the ellipsoid, and a specified polar triangle N
0
P
0
1
P
0
2
will, correspondingly, be on the
spherical surface, where N
0
P
0
1
ﾼ
90
o
u
1
, N
0
P
0
2
ﾼ
90
o
u
2
,
˃
is the great circle,
N
0
P
0
1
P
0
2
ﾼ
and
A
1
. Let the forward azimuth of the geodesic P
1
P
2
at the point P
2
be A
2
0
and the forward azimuth of the great circle P
0
1
P
0
2
at the point P
2
0
be
∠
ʱ
2
0
.
Applying the sine theorem to the spherical triangle N
0
P
0
1
P
0
2
results in:
0
2
cos u
1
sin A
1
ﾼ
cos u
2
sin
ʱ
:
ð
5
:
78
Þ
According to Clairaut's equation for geodesics, r sin A
ﾼ
C, and with (
5.73
)
r
ﾼ
x
ﾼ
a cos u, one obtains:
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