Editing Cylindrical coordinate system (section)

==Coordinate system conversions==
The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be used to convert between them.

===Cartesian coordinates===
For the conversion between cylindrical and Cartesian coordinates, it is convenient to assume that the reference plane of the former is the Cartesian {{mvar|xy}}-plane (with equation {{math|''z'' {{=}} 0}}), and the cylindrical axis is the Cartesian {{mvar|z}}-axis. Then the {{mvar|z}}-coordinate is the same in both systems, and the correspondence between cylindrical {{math|(''ρ'', ''φ'', ''z'')}} and Cartesian {{math|(''x'', ''y'', ''z'')}} are the same as for polar coordinates, namely
<math display="block">
\begin{align} 
x &= \rho \cos \varphi \\ 
y &= \rho \sin \varphi \\
z &= z
\end{align}
</math>
in one direction, and
<math display="block">\begin{align} \rho &= \sqrt{x^2+y^2} \\
\varphi &= \begin{cases}
\text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\
\arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\	
-\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\
-\arcsin\left(\frac{y}{\rho}\right) - \pi & \mbox{if } x < 0 \text{ and } y < 0
\end{cases} \end{align}</math>
in the other. The [[arcsine]] function is the inverse of the [[sine]] function, and is assumed to return an angle in the range {{math|[−{{sfrac|π|2}}, +{{sfrac|π|2}}]}} = {{math|[−90°, +90°]}}. These formulas yield an azimuth {{mvar|φ}} in the range {{math|[−180°, +180°]}}. 

By using the [[arctangent]] function that returns also an angle in the range {{math|[−{{sfrac|π|2}}, +{{sfrac|π|2}}]}} = {{math|[−90°, +90°]}}, one may also compute <math>\varphi</math> without computing <math>\rho</math> first
<math display="block">\begin{align}
\varphi &= \begin{cases}
\text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\
\frac\pi2\frac y{|y|} & \text{if } x = 0 \text{ and } y \ne 0\\
\arctan\left(\frac{y}{x}\right) & \mbox{if } x > 0 \\
\arctan\left(\frac{y}{x}\right)+\pi & \mbox{if } x < 0 \text{ and } y \ge 0\\
\arctan\left(\frac{y}{x}\right)-\pi & \mbox{if } x < 0 \text{ and } y < 0
\end{cases} \end{align}</math>
For other formulas, see the article [[Polar coordinate system]].

Many modern programming languages provide a function that will compute the correct azimuth {{mvar|φ}}, in the range {{math|(−π, π)}}, given ''x'' and ''y'', without the need to perform a case analysis as above.  For example, this function is called by {{mono|[[atan2]](''y'', ''x'')}} in the [[C (programming language)|C]] programming language, and {{mono|(atan ''y'' ''x'')}} in [[Common Lisp]].

===Spherical coordinates===
[[Spherical coordinates]] (radius {{mvar|r}}, elevation or inclination {{mvar|θ}}, azimuth {{mvar|φ}}), may be converted to or from cylindrical coordinates, depending on whether {{mvar|θ}} represents elevation or inclination, by the following:

{| class="wikitable plainrowheaders" style="text-align: center;"
|+ Conversion between spherical and cylindrical coordinates
|-
! scope="col" | Conversion to:
! scope="col" | Coordinate
! scope="col" | {{mvar|θ}} is elevation
! scope="col" | {{mvar|θ}} is inclination
|-
! scope="row" rowspan=3 style="font-weight: bold; text-align: center;" | Cylindrical
! scope="row" style="text-align: center;" | {{mvar|ρ}} =
| {{math|''r'' cos ''θ''}}
| {{math|''r'' sin ''θ''}}
|-
! scope="row" style="text-align: center;" | {{mvar|φ}} =
| colspan=  2 | {{mvar|φ}}
|-
! scope="row" style="text-align: center;" | {{mvar|z}} =
| {{math|''r'' sin ''θ''}}
| {{math|''r'' cos ''θ''}}
|-
! scope="row" rowspan=3 style="font-weight: bold; text-align: center;" | Spherical
! scope="row" style="text-align: center;" | {{mvar|r}} =
| colspan= 2  | <math display="inline">\sqrt{\rho^2+z^2}</math>
|-
! scope="row" style="text-align: center;" | {{mvar|θ}} =
| <math display="inline">\arctan\left(\frac{z}{\rho}\right)</math>
| <math display="inline">\arctan\left(\frac{\rho}{z}\right)</math>
|-
! scope="row" style="text-align: center;" | {{mvar|φ}} =
| colspan=  2 | {{mvar|φ}}
|}