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 angle between the direction and the equatorial plane, often called
*latitude*, say*b*, in the system under consideration (see examples below) - Each direction lies exactly in one plane perpendicular to the equatorial
plane, and including the polar axis. This plane cuts the equatorial one
in a well-determined line through the origin, and is uniquely described
by the angle from the preferred direction mentioned above, often called
*longitude*, say*l*, in the coordinate frame.

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