Editing Cylindrical coordinate system (section)

{{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>