The coordinates

Spherical coordinates use the three variables $(\rho, \phi, \theta)$. Here two of the coordinates are angles, while $\rho$ is the actual distance from the origin to the point.

The relationship between spherical and Cartesian is a little weird:

$$\rho = \sqrt{x^2 + y^2 + z^2}, \quad x = \rho \sin \phi \cos... ...\quad y = \rho \sin \phi \sin \theta, \quad z = \rho \cos \phi.$$

The constant coordinate surfaces are pretty intuitive. The surface $\rho = \rho_0$ describes all of the points at a distance $\rho_0$ from the origin. This is a sphere of radius $\rho_0$.

The surface $\phi = \phi_0$ is the strangest. It describes all the points that are at an angle of $\phi_0$ from the z-axis. This is a cone with an interior angle of $\phi_0$ with apex at the origin.

The surface $\theta = \theta_0$ describes a plane passing through the z-axis at an angle $\theta_0$ with the xaxis.