The coordinates

Cylindrical coordinates are a three dimensional version of polar coordinates. The variables here are $(r, \theta, z)$:

The relationship between cylindrical and Cartesian coordinates is

$$x = r \cos \theta, \quad y = r \sin \theta, \quad z = z,$$

to convert from cylindrical to Cartesian, and

$$r = \sqrt{x^2 + y^2}, \quad \theta = \arctan \left( \frac{y}{x} \right), \quad z = z,$$

to convert from Cartesian to cylindrical. Note that this makes r the distance from the z-axis.

Using this, we see that r = r₀ defines a cylinder of radius r₀ centered on the z-axis.

The equation $\theta = \theta_0$ defines a plane through the z-axis.

The equation z = z₀ defines a plane parallel to the xy-plane, passing through the point (0,0,z₀).