Spherical coordinate system can be considered as the three dimensional counterpart of polar coordinate system of two dimensional plane. While the polar coordinate system defines point with one angle and one distance, in spherical coordinate system, points are defined using two angles and one distance. In spherical coordinate system, as the name hints, all points in space are viewed to be placed on the surfaces of concentric spheres.

Any point P is referred using the coordinates ($\rho$, $\phi$, $\theta$) 
where, $\rho$ is the distance of the point from the origin O
$\phi$ is angle between line OP and positive z - axis
$\theta$ is the angle from the positive x - axis to the the projection of OP on XY plane.

The spherical coordinates are specially useful in solving problems, where we find symmetry about a point. The origin is shifted to this point and the equations are transformed to spherical system.

Spherical coordinates are also useful in dealing with some multiple integrals which cannot be evaluated using rectangular or cylindrical coordinates.

Spherical coordinates represent a point P in three dimensional space by ordered triples ($\rho$, $\phi$, $\theta$), in which
  1. $\rho$ is the distance of point P from the origin or $\rho$ = |$\overrightarrow{OP}$|
  2. $\phi$ is the angle $\overrightarrow{OP}$ makes with the positive z - axis.
  3. $\theta$ is the angle measured from the positive x - axis to $\overrightarrow{OR}$, the projection of $\overrightarrow{OP}$ on XY - plane.

Spherical Coordinates Definition


It can be observed from the above description that $\rho$ $\geq$ 0 and 0 $\leq$ $\theta$ $\leq$ $\pi$.

The relationships between the cartesian coordinates (x, y, z) to the corresponding spherical coordinates are as follows:
x = $\rho$ sin($\phi$) cos($\theta$)     
y = $\rho$ sin($\phi$) sin($\theta$)         
z = $\rho$ cos($\phi$)

Using distance formula, we also get
$\rho$2 = x2 + y2 + z2

Here, both $\phi$ and $\theta$ are measured in radians.