The cylindrical coordinate system is similar to that of the spherical coordinate system, but is an alternate extension to the polar coordinate system. Its elements, however, are something of a cross between the polar and Cartesian coordinates systems.

The coordinate system uses the standard polar coordinate system in the x-y plane, utilizing a distance from the origin (r) and an angle (θ) of extension from the positive x-axis (or pole). However, the third coordinate is a simple z-axis distance from above the x-y plane, just as any standard Cartesian system would utilize.

The coordinate ${\displaystyle (r, \theta, h)}$ represents the coordinate that exists at height h above the x-y plane (the z-coordinate). While, looking down from above, onto the x-y plane, the coordinate would appear to be at the polar coordinate (r, θ)

Conversion

Given the coordinates:

Spherical: ${\displaystyle (\rho, \phi, \theta)}$

Cylindrical: ${\displaystyle (r, \theta, h)}$

Cartesian: ${\displaystyle (x, y, z)}$

Spherical coordinates may be converted to cylindrical coordinates by:

${\displaystyle r = \rho \sin \phi \,}$

${\displaystyle \theta = \theta \,}$

${\displaystyle h = \rho \cos \phi \,}$

Cylindrical coordinates may be converted to spherical coordinates by:

${\displaystyle {\rho}=\sqrt{r^2+h^2}}$

${\displaystyle {\theta}=\theta \quad}$

${\displaystyle {\phi}=\arctan\frac{r}{h}}$

Cartesian coordinates may be converted into cylindrical by:

${\displaystyle r = \sqrt{x^2 + y^2}}$

${\displaystyle \theta = \arctan(\frac{y}{x})}$

${\displaystyle h = z}$

Cylindrical coordinates may be converted into Cartesian by:

${\displaystyle x = r \cos \theta}$

${\displaystyle y = r \sin \theta}$

${\displaystyle z = h}$