The local left-handed Cartesian coordinate system
can be defined by placing the origin to the local point
, whose z’-axis is pointed to the vertical,x’-axis is directed to the north, andy’ is pointed to the east (cf., Fig. 2.5). The x’y'-plane is called the horizontal plane; the vertical is defined perpendicular to the ellipsoid.
Fig. 2.5.
Astronomical coordinate system
Such a coordinate system is also called a local horizontal coordinate system. For any point
whose coordinates in the global and local coordinate system are
and
respectively, one has relations of
where A is the azimuth, Z is the zenith distance and d is the radius of the
in the local system. A is measured from the north clockwise; Z is the angle between the vertical and the radius d.
The local coordinate system
can indeed be obtained by two succeeded rotations of the global coordinate system (x, y, z) by _
and then by changing the x-axis to a right-handed system. In other words, the global system has to be rotated around the z-axis with angle
then around the y-axis with angle
and then change the sign of the x-axis. The total transformation matrix R is then
and there are:
where
are the same vector represented in local and global coordinate systems.
are the geodetic latitude and longitude of the local point.
If the vertical direction is defined as the plump line of the gravitational field at the local point, then such a local coordinate system is called an astronomic horizontal system (its x’-axis is pointed to the north, left-handed system). The plump line of gravity g and the vertical line of the ellipsoid at the point p are generally not coinciding with each other; however, the difference is very small. The difference is omitted in GPS practice.
Combining Eqs. 2.10 and 2.12, the zenith angle and azimuth of a point P2 (satellite) related to the station P1 can be directly computed by using the global coordinates of the two points by
where
