Spherical Co-ordinates

It turns out to be very convenient for the problem solving in cartography to operate with the so called spherical co-ordinates instead of cartesian co-ordinates (figure from [3] under literature).

 parameter representation of the sphere

Thus any point P in space can be determined by the number triple (R; v; u) instead of the number triple (x; y; z). Here it is:

R = radius of sphere = distance between the point P and the origin O ( 0 ≤ R ),

=  angle between the line OP and the xy-plane ( -  / 2 ≤ ≤  / 2 ) and

=  angle between the projection of the line OP onto the xy-plane and the positive x-axis ( -  ≤ ≤  ).

The illustration shows the rotational direction of the goniometry. The values R, v, u are named spherical co-ordinates of the point P. They correspond to the polar co-ordinates in the plane and therefore are also called spacial polar co-ordinates.

Any triple of spherical co-ordinates accords to exactly one point in space. Contrariwise a point P in space accords to a single triple of spherical co-ordinates only if P does not lie onto the z-axis: On the z-axis without the origin O only R and (±  / 2) are unique, in contrast u  is arbitrarily. If P is equal to the origin O, thus only R = 0 is unique, v and u are arbitrary.

  
See also
  
 parameter representation parameter representation
 parameter lines parameter lines