Cylindrical coordinate system

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 $(r, \theta, h)$ 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: $(\rho, \phi, \theta)$

Cylindrical: $(r, \theta, h)$

Cartesian: $(x, y, z)$

Spherical coordinates may be converted to cylindrical coordinates by:

$r = \rho \sin \phi \,$

$\theta = \theta \,$

$h = \rho \cos \phi \,$

Cylindrical coordinates may be converted to spherical coordinates by:

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

${\theta}=\theta \quad$

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

Cartesian coordinates may be converted into cylindrical by:

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

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

$h = z$

Cylindrical coordinates may be converted into Cartesian by:

$x = r \cos \theta$

$y = r \sin \theta$

$z = h$