Editing Cylindrical coordinate system

{{Use American English|date = March 2019}}
{{Short description|Coordinates comprising two distances and an angle}}
[[Image:Coord system CY 1.svg|thumb|240px|A cylindrical coordinate system with origin {{mvar|O}}, polar axis {{mvar|A}}, and longitudinal axis {{mvar|L}}. The dot is the point with radial distance {{math|''ρ'' {{=}} 4}}, angular coordinate {{math|''φ'' {{=}} 130°}}, and height {{math|''z'' {{=}} 4}}. The reference plane contains the purple section.]]

A '''cylindrical coordinate system''' is a [[three-dimensional]] [[coordinate system]] that specifies point positions around a main axis (a chosen [[directed line]]) and an auxiliary axis (a reference [[ray (geometry)|ray]]). The three cylindrical [[coordinates]] are: the point [[perpendicular distance]] {{math|''ρ''}} from the main axis; the point [[signed distance]] ''z'' along the main axis from a chosen [[origin (mathematics)|origin]]; and the [[plane angle]] {{math|''φ''}} of the [[Vector projection on a plane|point projection]] on a reference plane (passing through the origin and perpendicular to the main axis) 

The main axis is variously called the ''cylindrical'' or ''longitudinal'' axis. The auxiliary axis is called the ''polar axis'', which lies in the reference plane, starting at the origin, and pointing in the reference direction.
Other directions perpendicular to the longitudinal axis are called ''radial lines''.

The distance from the axis may be called the ''radial distance'' or ''radius'', while the angular coordinate is sometimes referred to as the ''angular position'' or as the ''azimuth''. The radius and the azimuth are together called the ''polar coordinates'', as they correspond to a two-dimensional [[polar coordinates|polar coordinate]] system in the plane through the point, parallel to the reference plane. The third coordinate may be called the ''height'' or ''altitude'' (if the reference plane is considered horizontal), ''longitudinal position'',<ref>{{cite journal |last1=Krafft |first1=C. |last2=Volokitin |first2=A. S. |title=Resonant electron beam interaction with several lower hybrid waves |journal=Physics of Plasmas |date=1 January 2002 |volume=9 |issue=6 |pages=2786–2797 |doi=10.1063/1.1465420 |url=http://pop.aip.org/resource/1/phpaen/v9/i6/p2786_s1?isAuthorized=no |access-date=9 February 2013 |issn=1089-7674 |quote=...in cylindrical coordinates {{math|(''r'',''θ'',''z'')}} ... and {{math|''Z'' {{=}} ''v<sub>bz</sub>t''}} is the longitudinal position... |bibcode=2002PhPl....9.2786K |archive-url=https://archive.today/20130414005110/http://pop.aip.org/resource/1/phpaen/v9/i6/p2786_s1?isAuthorized=no |archive-date=14 April 2013 |url-status=dead }}</ref> or ''axial position''.<ref>{{cite journal | last1 = Groisman | first1 = Alexander | last2 = Steinberg | first2 = Victor | year = 1997 | title = Solitary Vortex Pairs in Viscoelastic Couette Flow | journal = Physical Review Letters | volume = 78 | issue = 8| pages = 1460–1463 | doi = 10.1103/PhysRevLett.78.1460 | bibcode=1997PhRvL..78.1460G |quote=...where {{mvar|r}}, {{mvar|θ}}, and {{mvar|z}} are cylindrical coordinates ... as a function of axial position...| arxiv = patt-sol/9610008 | s2cid = 54814721 }}</ref>

Cylindrical coordinates are useful in connection with objects and phenomena that have some rotational [[symmetry]] about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal [[cylinder (geometry)|cylinder]], [[electromagnetic fields]] produced by an [[electric current]] in a long, straight wire, [[accretion disk]]s in astronomy, and so on.

They are sometimes called ''cylindrical polar coordinates''<ref>{{cite book|first=J. E. |last=Szymanski |title=Basic Mathematics for Electronic Engineers: models and applications |series=Tutorial Guides in Electronic Engineering (no. 16) |publisher=Taylor & Francis |date=1989 |isbn=978-0-278-00068-1 |url=https://books.google.com/books?id=L7wOAAAAQAAJ&q=%22Cylindrical+polar+coordinate%22&pg=PA170 |page=170}}</ref> or ''polar cylindrical coordinates'',<ref>{{cite book|first=Robert H.|last= Nunn |title=Intermediate Fluid Mechanics |publisher=Taylor & Francis |date=1989 |isbn=978-0-89116-647-4 |url=https://books.google.com/books?id=0KfkkbX-NYQC&q=%22polar+Cylindrical++coordinate%22&pg=PA3 |page=3}}</ref> and are sometimes used to specify the position of stars in a galaxy (''galactocentric cylindrical polar coordinates'').<ref>{{cite book|first1=Linda Siobhan |last1=Sparke |author1-link=Linda Sparke|first2=John Sill |last2=Gallagher |title=Galaxies in the Universe: An Introduction |edition=2nd |publisher=Cambridge University Press |date=2007 |isbn=978-0-521-85593-8 |url=https://books.google.com/books?id=N8Hngab5liQC&q=cylindrical+polar+coordinate+galaxy&pg=PA37 |page=37}}</ref>

==Definition==
The three coordinates ({{mvar|[[Rho (letter)|ρ]]}}, {{mvar|[[Phi|φ]]}}, {{mvar|z}}) of a point {{mvar|P}} are defined as:
* The ''radial distance'' {{mvar|ρ}} is the [[Euclidean distance]] from the {{mvar|z}}-axis to the point {{mvar|P}}.
* The ''azimuth'' {{mvar|φ}} is the angle between the reference direction on the chosen plane and the line from the origin to the projection of {{mvar|P}} on the plane.
* The ''axial coordinate'' or ''height'' {{mvar|z}} is the signed distance from the chosen plane to the point {{mvar|P}}.

===Unique cylindrical coordinates===
As in polar coordinates, the same point with cylindrical coordinates {{math|(''ρ'', ''φ'', ''z'')}} has infinitely many equivalent coordinates, namely {{math|(''ρ'', ''φ'' ± ''n''×360°, ''z'')}} and {{math|(−''ρ'', ''φ'' ± (2''n'' + 1)×180°, ''z''),}} where {{mvar|n}} is any integer. Moreover, if the radius {{mvar|ρ}} is zero, the azimuth is arbitrary.

In situations where someone wants a unique set of coordinates for each point, one may restrict the radius to be [[non-negative]] ({{math|''ρ'' ≥ 0}}) and the azimuth {{mvar|φ}} to lie in a specific [[interval (mathematics)|interval]] spanning 360°, such as {{math|[−180°,+180°]}} or {{math|[0,360°]}}.

===Conventions===
The notation for cylindrical coordinates is not uniform. The [[International Organization for Standardization|ISO]] standard [[ISO 31-11|31-11]] recommends {{math|(''ρ'', ''φ'', ''z'')}}, where {{mvar|ρ}} is the radial coordinate, {{mvar|φ}} the azimuth, and {{mvar|z}} the height. However, the radius is also often denoted {{mvar|r}} or {{mvar|s}}, the azimuth by {{mvar|θ}} or {{mvar|t}}, and the third coordinate by {{mvar|h}} or (if the cylindrical axis is considered horizontal) {{mvar|x}}, or any context-specific letter.

[[File:Cylindrical coordinate surfaces.png|thumb|right|The [[coordinate surfaces]] of the cylindrical coordinates {{math|(''ρ'', ''φ'', ''z'')}}. The red [[Cylinder (geometry)|cylinder]] shows the points with {{math|''ρ'' {{=}} 2}}, the blue [[plane (mathematics)|plane]] shows the points with {{math|''z'' {{=}} 1}}, and the yellow half-plane shows the points with {{math|''φ'' {{=}} −60°}}. The {{mvar|z}}-axis is vertical and the {{mvar|x}}-axis is highlighted in green.  The three surfaces intersect at the point {{mvar|P}} with those coordinates (shown as a black sphere); the [[Cartesian coordinate system|Cartesian coordinates]] of {{mvar|P}} are roughly (1.0, −1.732, 1.0).]]

[[File:Cylindrical coordinate surfaces.gif|thumb|Cylindrical coordinate surfaces. The three orthogonal components, {{mvar|ρ}} (green), {{mvar|φ}} (red), and {{mvar|z}} (blue), each increasing at a constant rate. The point is at the intersection between the three colored surfaces.]]

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured [[clockwise|counterclockwise]] as seen from any point with positive height.

==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}}
|}

==Line and volume elements==
{{hatnote|1= See [[Multiple integral#Cylindrical coordinates|multiple integral]] for details of volume integration in cylindrical coordinates, and [[Del in cylindrical and spherical coordinates]] for [[vector calculus]] formulae.}}

In many problems involving cylindrical polar coordinates, it is useful to know the line and volume elements; these are used in integration to solve problems involving paths and volumes.

The [[line element]] is
<math display="block">\mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}.</math>

The [[volume element]] is
<math display="block">\mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>

The [[Differential (infinitesimal)|surface element]] in a surface of constant radius {{mvar|ρ}} (a vertical cylinder) is
<math display="block">\mathrm{d}S_\rho = \rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>

The surface element in a surface of constant azimuth {{mvar|φ}} (a vertical half-plane) is
<math display="block">\mathrm{d}S_\varphi = \mathrm{d}\rho\,\mathrm{d}z.</math>

The surface element in a surface of constant height {{mvar|z}} (a horizontal plane) is
<math display="block">\mathrm{d}S_z = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.</math>

The [[del]] operator in this system leads to the following expressions for [[gradient]], [[divergence]], [[curl (mathematics)|curl]] and [[Laplacian]]:
<math display="block">\begin{align}
  \nabla f &= \frac{\partial f}{\partial \rho}\boldsymbol{\hat{\rho}} + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\boldsymbol{\hat{\varphi}} + \frac{\partial f}{\partial z}\boldsymbol{\hat{z}} \\[8px]

  \nabla \cdot \boldsymbol{A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_\rho\right) + \frac{1}{\rho} \frac{\partial A_\varphi}{\partial \varphi} + \frac{\partial A_z}{\partial z} \\[8px]

  \nabla \times \boldsymbol{A} &= \left(\frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z}\right)\boldsymbol{\hat{\rho}} +
    \left(\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho}\right)\boldsymbol{\hat{\varphi}} +
    \frac{1}{\rho}\left(\frac{\partial}{\partial \rho}\left(\rho A_\varphi\right) - \frac{\partial A_\rho}{\partial \varphi}\right) \boldsymbol{\hat{z}} \\[8px]

  \nabla^2 f &= \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho}\right) +
    \frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2}
\end{align}</math>

==Cylindrical harmonics==
The solutions to the [[Laplace equation]] in a system with cylindrical symmetry are called [[cylindrical harmonics]].

==Kinematics==
In a cylindrical coordinate system, the position of a particle can be written as<ref name="Taylor">{{cite book |last1=Taylor |first1=John R. |title=Classical Mechanics |date=2005 |publisher=University Science Books |location=Sausalito, California |page=29}}</ref>
<math display="block">\boldsymbol{r} = \rho\,\boldsymbol{\hat \rho} + z\,\boldsymbol{\hat z}.</math>
The velocity of the particle is the time derivative of its position,
<math display="block">\boldsymbol{v} = \frac{\mathrm{d}\boldsymbol{r}}{\mathrm{d}t} = \dot{\rho}\,\boldsymbol{\hat \rho} + \rho\,\dot\varphi\,\hat{\boldsymbol{\varphi}} + \dot{z}\,\hat{\boldsymbol{z}},</math>
where the term <math>\rho \dot\varphi\hat\varphi</math> comes from the Poisson formula <math>\frac{\mathrm d\hat\rho}{\mathrm dt} = \dot\varphi\hat z\times \hat\rho </math>. Its acceleration is<ref name="Taylor"/>
<math display="block">
\boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } +  \ddot{z}\,\hat{\boldsymbol{z}}
</math>

==See also==
*[[List of canonical coordinate transformations]]
*[[Vector fields in cylindrical and spherical coordinates]]
*[[Del in cylindrical and spherical coordinates]]

==References==
{{reflist}}

==Further reading==
*{{cite book |last1=Morse |first1=Philip M. |author-link1=Philip M. Morse |last2=Feshbach |first2=Herman |author-link2=Herman Feshbach |year=1953 |title=Methods of Theoretical Physics, Part I |publisher=[[McGraw-Hill]] |location=[[New York City]] |isbn=0-07-043316-X |lccn=52011515 |pages=656–657}}
*{{cite book |last1=Margenau |first1=Henry |author-link1=Henry Margenau |last2=Murphy |first2=George M. |year=1956 |title=The Mathematics of Physics and Chemistry |url=https://archive.org/details/mathematicsofphy0002marg |url-access=registration |publisher=D. van Nostrand |location=New York City |page=[https://archive.org/details/mathematicsofphy0002marg/page/178 178] |lccn=55010911 |isbn=9780882754239 |oclc=3017486}}
*{{cite book |last1=Korn |first1=Granino A. |last2=Korn |first2=Theresa M. |author2-link= Theresa M. Korn |year=1961 |title=Mathematical Handbook for Scientists and Engineers |url=https://archive.org/details/mathematicalhand0000korn |url-access=registration |publisher=McGraw-Hill |location=New York City |id=ASIN B0000CKZX7 | pages=[https://archive.org/details/mathematicalhand0000korn/page/174 174–175] | lccn=59014456}}
*{{cite book |last1=Sauer |first1=Robert |last2=Szabó |first2=István |year=1967 |title=Mathematische Hilfsmittel des Ingenieurs |publisher=[[Springer Science+Business Media|Springer-Verlag]] |location=New York City |page=95 |lccn=67025285}}
*{{cite book |last=Zwillinger |first=Daniel |year=1992 |title=Handbook of Integration |publisher=[[Jones and Bartlett Publishers]] |location=[[Boston]] |isbn=0-86720-293-9 |page=113 |oclc=25710023}}
*{{cite book |last1=Moon |first1=P. |last2=Spencer |first2=D. E. |year=1988 |chapter=Circular-Cylinder Coordinates (r, ψ, z) |title=Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions |edition=corrected 2nd |publisher=Springer-Verlag |location=New York City |pages=12–17, Table 1.02 |isbn=978-0-387-18430-2}}

==External links==
* {{springer|title=Cylinder coordinates|id=p/c027600}}
*[http://mathworld.wolfram.com/CylindricalCoordinates.html MathWorld description of cylindrical coordinates]
*[https://web.archive.org/web/20100708120521/http://www.math.montana.edu/frankw/ccp/multiworld/multipleIVP/cylindrical/body.htm Cylindrical Coordinates]  Animations illustrating cylindrical coordinates by Frank Wattenberg

{{Orthogonal coordinate systems}}

[[Category:Three-dimensional coordinate systems]]
[[Category:Orthogonal coordinate systems]]

[[de:Polarkoordinaten#Zylinderkoordinaten]]
[[ro:Coordonate polare#Coordonate cilindrice]]
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