Spherical coordinate systems

In many cases, the Cartesian coordinate system is not the most convenient one to describe the position. For example, when we should like to define a position on the surface of a sphere, it is more convenient the express the position with certain angles. On the surface of the Earth, we use longitudes and latitudes to specify the position of a place.

The reference objects in a spherical coordinate system are

• a fixed point: the origin;
• a fixed plane containing the origin;
• a fixed direction, represented by a vector in this plane whose tail is the origin.

Using these, we can specify the coordinates of a point P:

• radial distance of P from the origin: \(r;\)
• elevation angle measured from the fixed plane to P: \(\vartheta;\)
• azimuth angle --- the angle between the reference direction and the orthogonal projection of P on the fixed plane: \(\phi.\)

In two dimensions, coordinates are reduced to

• radial distance \(r\) from the origin;
• angle \(\phi\) between the position vector and the reference vector.