Geoscience Reference
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5.6 Relationship Between the Geodetic Coordinate System
and the Geodesic Polar Coordinate System
5.6.1 Geodesic Polar Coordinate Systems and the Solution
of Geodetic Problems
A geodesic polar coordinate system is established on the surface of the ellipsoid.
The position of a point on the ellipsoid is represented by the geodesic distance S and
geodetic azimuth A from the polar point to this point of interest, as shown in
Fig.
5.39
. Let P
1
be the polar point on the ellipsoid, the meridian P
1
N that passes
through P
1
be the polar axis, the geodesic distance S that connects P
1
and P (to be
computed) be the polar radius, and the geodetic azimuth A of the geodesic at point
P
1
be the polar angle; then the location of point P on the ellipsoid is expressed by (S,
A).
The geodesic polar coordinate system is used to show the relative horizontal
positions between two points on the ellipsoid, often applied in the case where
solution of relative positions is needed for long-distance weapon launching or
navigation.
Calculations of geodetic coordinates of an unknown point on the ellipsoid based
on the observed angles and distances using geodetic surveying or calculations of the
geodesic distance and geodetic azimuth between two points based on their geodetic
coordinates are known as solutions of geodetic problems, calculations of geodetic
coordinates, or calculations of geodetic positions. The solutions of geodetic prob-
lems include both direct and inverse solutions.
In Fig.
5.40
, given the geodetic coordinates (L
1,
B
1
) of point P
1
, the geodesic
distance S from point P
1
to P
2
, and the geodetic azimuth A
1
from P
1
to P
2
, the direct
solution of the geodetic problem provides the geodetic coordinates (L
2
,B
2
) of point
P
2
and the reverse azimuth A
2
of the geodesic at point P
2
. Given the geodetic
coordinates (L
1,
B
1
) and (L
2
,B
2
)ofP
1
and P
2
, the inverse solution of the geodetic
problem is required to find the forward and reverse azimuths A
1
, A
2
and the
geodesic distance S of P
1
and P
2
. From the definition of the geodesic polar
coordinates, (S, A
1
) and (S, A
2
) are the geodesic polar coordinates of points P
2
and P
1
, respectively. Hence, the solution of geodetic problems is the interconver-
sion between the geodetic coordinates and geodesic polar coordinates.
The solution of geodetic problems can be applied in many ways. It can be used to
calculate the geodetic coordinates on the ellipsoid in geodetic surveying (as shown
in Fig.
5.29
). Apart from that, with the advancement of modern spatial technology,
aviation, and navigation, the solution of geodetic problems (particularly the inverse
solution of geodetic problems) is playing a more prominent role. These different
applications and requirements also generate different methods and formulae for the
solution of geodetic problems.
Computing geodetic coordinates on the ellipsoid is far more complicated than
computing coordinates on the plane due to the significantly more complex
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