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General Coordinate Systems

For identifying each position in space uniquely, three numbers are required, as space is three-dimensional (these numbers correspond to length, width, and height, for example, as they teach you in school).

If the numbers are assigned in such a way that neighboring positions are assigned with neighboring numbers (what mathematicians call "continuously") and in a unique way (i.e., each set of numbers corresponds to exactly one position in space), this system of assigned numbers is called a coordinate system. From what is stated, it is obvious that there is an infinite number of possibilities to introduce a coordinate system in space. In practice, however, astronomy and other sciences prefer to deal with certain specific systems, which have been described above. The numbers assigned to a position, or point, are called the coordinates of that point (in the coordinate system under consideration).

In physical laboratories, and sometimes for everyday purpose, it is convenient to use cartesian coordinates, i.e., just length, width and height measured from a reference point (the origin of the system) in three mutually othogonal (or perpendicular) directions, i.e., three straight lines called axes of the coordinate system, which have mutually right angles between them. Traditionally, these axes are given a sense of orientation, and labelled "X", "Y", and "Z", so that the positive X, Y, and Z axis form a right handed rectangular frame. Both the origin and the axes can be freely chosen in space, but then kept fixed in the coordinate system under consideration. For the frame chosen, each two of the axes lie in a plane, and the three coordinate planes formed this way are also characteristic for the system; these are sometimes called the X-Y, X-Z, and Y-Z plane.

If the position of an object can be measured at same precession in each direction, the cartesian coordinate system is quite practical, e.g. in many laboratories. In astronomy, however, we have the situation that while the direction from the observer, Earth, or even the Solar System to an object can be measured with good accuracy, the distance is often poorly known.

Directions, given by straight half lines starting at the origin, are best measured by angles to given reference axes or reference frames. A position in space can be uniquely determined by its distance and its direction from a reference point, e.g. the origin of a given reference frame. In space, the direction is given by two angles, so that together with the distance, again three numbers determine a position. These three number can be taken as a coordinate system, which is then called a spherical coordinate system.

For each cartesian coordinate system, one can find a spherical coordinate system by the following conventions: Call the Z axis the Polar Axis, the positive Z direction the North Pole and the negative Z axis the South Pole of the coordinate system, and the X-Y plane (which is perpendicular to the polar axis) the Equatorial Plane of the coordinate system. In order to characterize any direction uniquely, one needs an additional preferred direction within the equatorial plane, e.g. the X axis ot the cartesian coordinate system. Each coordinate system is now uniquely determined by its origin, either its polar axis or the equatorial plane (the other is always perpendicular and thus given by the one), and the reference direction. Then, any direction can be uniquely characterized by

The spherical coordinates of any location in a given reference frame can be obtaind from the cartesian ones by the following transformation formulae, with the distance called R:
    X = R * cos b * cos l
    Y = R * cos b * sin l
    Z = R * sin b

    R^2 = X^2 + Y^2 + Z^2
    tan l = Y/X
    sin b = Z/R

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