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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.

Cylindrical coordinates

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 $
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