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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.

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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