Cylindrical coordinate system

Acylindrical coordinate systemis a three-dimensionalcoordinate systemthat specifies point positions by the distance from a chosen reference axis(axis L in the image opposite),the direction from the axis relative to a chosen reference direction(axis A),and the distance from a chosen reference plane perpendicular to the axis(plane containing the purple section).The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

Theoriginof the system is the point where all three coordinates can be given as zero. This is the intersection between the reference plane and the axis. The axis is variously called thecylindricalorlongitudinalaxis, to differentiate it from thepolar axis,which is theraythat lies in the reference plane, starting at the origin and pointing in the reference direction. Other directions perpendicular to the longitudinal axis are calledradial lines.

The distance from the axis may be called theradial distanceorradius,while the angular coordinate is sometimes referred to as theangular positionor as theazimuth.The radius and the azimuth are together called thepolar coordinates,as they correspond to a two-dimensionalpolar coordinatesystem in the plane through the point, parallel to the reference plane. The third coordinate may be called theheightoraltitude(if the reference plane is considered horizontal),longitudinal position,^[1]oraxial position.^[2]

Cylindrical coordinates are useful in connection with objects and phenomena that have some rotationalsymmetryabout the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metalcylinder,electromagnetic fieldsproduced by anelectric currentin a long, straight wire,accretion disksin astronomy, and so on.

They are sometimes called "cylindrical polar coordinates"^[3]and "polar cylindrical coordinates",^[4]and are sometimes used to specify the position of stars in a galaxy ( "galactocentric cylindrical polar coordinates" ).^[5]

Definition

The three coordinates ( $ρ$ , $φ$ , $z$ ) of a point $P$ are defined as:

Theradial distance $ρ$ is theEuclidean distancefrom the $z$ -axis to the point $P$ .
Theazimuth $φ$ is the angle between the reference direction on the chosen plane and the line from the origin to the projection of $P$ on the plane.
Theaxial coordinateorheight $z$ is the signed distance from the chosen plane to the point $P$ .

Unique cylindrical coordinates

As in polar coordinates, the same point with cylindrical coordinates $(ρ, φ, z)$ has infinitely many equivalent coordinates, namely $(ρ, φ \pm n \times360°, z)$ and $(- ρ, φ \pm (2 n + 1)\times180°, z),$ where $n$ is any integer. Moreover, if the radius $ρ$ 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 benon-negative( $ρ \geq 0$ ) and the azimuth $φ$ to lie in a specificintervalspanning 360°, such as $[-180°,+180°]$ or $[0,360°]$ .

Conventions

The notation for cylindrical coordinates is not uniform. TheISOstandard31-11recommends $(ρ, φ, z)$ ,where $ρ$ is the radial coordinate, $φ$ the azimuth, and $z$ the height. However, the radius is also often denoted $r$ or $s$ ,the azimuth by $θ$ or $t$ ,and the third coordinate by $h$ or (if the cylindrical axis is considered horizontal) $x$ ,or any context-specific letter.

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measuredcounterclockwiseas 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 $xy$ -plane (with equation $z = 0$ ), and the cylindrical axis is the Cartesian $z$ -axis. Then the $z$ -coordinate is the same in both systems, and the correspondence between cylindrical $(ρ, φ, z)$ and Cartesian $(x, y, z)$ are the same as for polar coordinates, namely ${\begin{aligned}x&=\rho \cos \varphi \\y&=\rho \sin \varphi \\z&=z\end{aligned}}$ in one direction, and ${\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\varphi &={\begin{cases}{\text{indeterminate}}&{\text{if }}x=0{\text{ and }}y=0\\\arcsin \left({\frac {y}{\rho }}\right)&{\text{if }}x\geq 0\\-\arcsin \left({\frac {y}{\rho }}\right)+\pi &{\mbox{if }}x<0{\text{ and }}y\geq 0\\-\arcsin \left({\frac {y}{\rho }}\right)+\pi &{\mbox{if }}x<0{\text{ and }}y<0\end{cases}}\end{aligned}}$ in the other. Thearcsinefunction is the inverse of thesinefunction, and is assumed to return an angle in the range $[−.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠π/2⁠,+⁠π/2⁠]$ = $[-90°, +90°]$ .These formulas yield an azimuth $φ$ in the range $[-90°, +270°]$ .

By using thearctangentfunction that returns also an angle in the range $[- ⁠ π / 2 ⁠,+ ⁠ π / 2 ⁠]$ = $[-90°, +90°]$ ,one may also compute $\varphi$ without computing $\rho$ first ${\begin{aligned}\varphi &={\begin{cases}{\text{indeterminate}}&{\text{if }}x=0{\text{ and }}y=0\\{\frac {\pi }{2}}{\frac {y}{|y|}}&{\text{if }}x=0{\text{ and }}y\neq 0\\\arctan \left({\frac {y}{x}}\right)&{\mbox{if }}x>0\\\arctan \left({\frac {y}{x}}\right)+\pi &{\mbox{if }}x<0{\text{ and }}y\geq 0\\\arctan \left({\frac {y}{x}}\right)-\pi &{\mbox{if }}x<0{\text{ and }}y<0\end{cases}}\end{aligned}}$ For other formulas, see the articlePolar coordinate system.

Many modern programming languages provide a function that will compute the correct azimuth $φ$ ,in the range $(-π, π)$ ,givenxandy,without the need to perform a case analysis as above. For example, this function is called byatan2(y,x)in theCprogramming language, and(atanyx)inCommon Lisp.

Spherical coordinates

Spherical coordinates(radius $r$ ,elevation or inclination $θ$ ,azimuth $φ$ ), may be converted to or from cylindrical coordinates, depending on whether $θ$ represents elevation or inclination, by the following:

Conversion between spherical and cylindrical coordinates
Conversion to:	Coordinate	$θ$ is elevation	$θ$ is inclination
Cylindrical	$ρ$ =	$r cos θ$	$r sin θ$
	$φ$ =	$φ$
	$z$ =	$r sin θ$	$r cos θ$
Spherical	$r$ =	${\textstyle {\sqrt {\rho ^{2}+z^{2}}}}$
	$θ$ =	${\textstyle \arctan \left({\frac {z}{\rho }}\right)}$	${\textstyle \arctan \left({\frac {\rho }{z}}\right)}$
	$φ$ =	$φ$

Line and volume elements

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.

Theline elementis $\mathrm {d} {\boldsymbol {r}}=\mathrm {d} \rho \,{\boldsymbol {\hat {\rho }}}+\rho \,\mathrm {d} \varphi \,{\boldsymbol {\hat {\varphi }}}+\mathrm {d} z\,{\boldsymbol {\hat {z}}}.$

Thevolume elementis $\mathrm {d} V=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi \,\mathrm {d} z.$

Thesurface elementin a surface of constant radius $ρ$ (a vertical cylinder) is $\mathrm {d} S_{\rho }=\rho \,\mathrm {d} \varphi \,\mathrm {d} z.$

The surface element in a surface of constant azimuth $φ$ (a vertical half-plane) is $\mathrm {d} S_{\varphi }=\mathrm {d} \rho \,\mathrm {d} z.$

The surface element in a surface of constant height $z$ (a horizontal plane) is $\mathrm {d} S_{z}=\rho \,\mathrm {d} \rho \,\mathrm {d} \varphi.$

Thedeloperator in this system leads to the following expressions forgradient,divergence,curlandLaplacian: ${\begin{aligned}\nabla f&={\frac {\partial f}{\partial \rho }}{\boldsymbol {\hat {\rho }}}+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}{\boldsymbol {\hat {\varphi }}}+{\frac {\partial f}{\partial z}}{\boldsymbol {\hat {z}}}\\[8px]\nabla \cdot {\boldsymbol {A}}&={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho A_{\rho }\right)+{\frac {1}{\rho }}{\frac {\partial A_{\varphi }}{\partial \varphi }}+{\frac {\partial A_{z}}{\partial z}}\\[8px]\nabla \times {\boldsymbol {A}}&=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\boldsymbol {\hat {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\boldsymbol {\hat {\varphi }}}+{\frac {1}{\rho }}\left({\frac {\partial }{\partial \rho }}\left(\rho A_{\varphi }\right)-{\frac {\partial A_{\rho }}{\partial \varphi }}\right){\boldsymbol {\hat {z}}}\\[8px]\nabla ^{2}f&={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}\end{aligned}}$

Cylindrical harmonics

The solutions to theLaplace equationin a system with cylindrical symmetry are calledcylindrical harmonics.

Kinematics

In a cylindrical coordinate system, the position of a particle can be written as^[6] ${\boldsymbol {r}}=\rho \,{\boldsymbol {\hat {\rho }}}+z\,{\boldsymbol {\hat {z}}}.$ The velocity of the particle is the time derivative of its position, ${\boldsymbol {v}}={\frac {\mathrm {d} {\boldsymbol {r}}}{\mathrm {d} t}}={\dot {\rho }}\,{\boldsymbol {\hat {\rho }}}+\rho \,{\dot {\varphi }}\,{\hat {\boldsymbol {\varphi }}}+{\dot {z}}\,{\hat {\boldsymbol {z}}},$ where the term $\rho {\dot {\varphi }}{\hat {\varphi }}$ comes from the Poisson formula ${\frac {\mathrm {d} {\hat {\rho }}}{\mathrm {d} t}}={\dot {\varphi }}{\hat {z}}\times {\hat {\rho }}$ .Its acceleration is^[6] ${\boldsymbol {a}}={\frac {\mathrm {d} {\boldsymbol {v}}}{\mathrm {d} t}}=\left({\ddot {\rho }}-\rho \,{\dot {\varphi }}^{2}\right){\boldsymbol {\hat {\rho }}}+\left(2{\dot {\rho }}\,{\dot {\varphi }}+\rho \,{\ddot {\varphi }}\right){\hat {\boldsymbol {\varphi }}}+{\ddot {z}}\,{\hat {\boldsymbol {z}}}$

References

^Krafft, C.; Volokitin, A. S. (1 January 2002)."Resonant electron beam interaction with several lower hybrid waves".Physics of Plasmas.9(6): 2786–2797.Bibcode:2002PhPl....9.2786K.doi:10.1063/1.1465420.ISSN 1089-7674.Archived fromthe originalon 14 April 2013.Retrieved9 February2013....in cylindrical coordinates $(r, θ, z)$ ... and $Z = v bz t$ is the longitudinal position...
^Groisman, Alexander; Steinberg, Victor (1997). "Solitary Vortex Pairs in Viscoelastic Couette Flow".Physical Review Letters.78(8): 1460–1463.arXiv:patt-sol/9610008.Bibcode:1997PhRvL..78.1460G.doi:10.1103/PhysRevLett.78.1460.S2CID 54814721....where $r$ , $θ$ ,and $z$ are cylindrical coordinates... as a function of axial position...
^Szymanski, J. E. (1989).Basic Mathematics for Electronic Engineers: models and applications.Tutorial Guides in Electronic Engineering (no. 16). Taylor & Francis. p. 170.ISBN 978-0-278-00068-1.
^Nunn, Robert H. (1989).Intermediate Fluid Mechanics.Taylor & Francis. p. 3.ISBN 978-0-89116-647-4.
^Sparke, Linda Siobhan;Gallagher, John Sill (2007).Galaxies in the Universe: An Introduction(2nd ed.). Cambridge University Press. p. 37.ISBN 978-0-521-85593-8.
^^a ^bTaylor, John R. (2005).Classical Mechanics.Sausalito, California: University Science Books. p. 29.

External links

"Cylinder coordinates",Encyclopedia of Mathematics,EMS Press,2001 [1994]
MathWorld description of cylindrical coordinates
Cylindrical CoordinatesAnimations illustrating cylindrical coordinates by Frank Wattenberg

[1] Krafft, C.; Volokitin, A. S. (1 January 2002)."Resonant electron beam interaction with several lower hybrid waves".Physics of Plasmas.9(6): 2786–2797.Bibcode:2002PhPl....9.2786K.doi:10.1063/1.1465420.ISSN 1089-7674.Archived fromthe originalon 14 April 2013.Retrieved9 February2013....in cylindrical coordinates $(r, θ, z)$ ... and $Z = v bz t$ is the longitudinal position...

[2] Groisman, Alexander; Steinberg, Victor (1997). "Solitary Vortex Pairs in Viscoelastic Couette Flow".Physical Review Letters.78(8): 1460–1463.arXiv:patt-sol/9610008.Bibcode:1997PhRvL..78.1460G.doi:10.1103/PhysRevLett.78.1460.S2CID 54814721....where $r$ , $θ$ ,and $z$ are cylindrical coordinates... as a function of axial position...

[3] Szymanski, J. E. (1989).Basic Mathematics for Electronic Engineers: models and applications.Tutorial Guides in Electronic Engineering (no. 16). Taylor & Francis. p. 170.ISBN 978-0-278-00068-1.

[4] Nunn, Robert H. (1989).Intermediate Fluid Mechanics.Taylor & Francis. p. 3.ISBN 978-0-89116-647-4.

[5] Sparke, Linda Siobhan;Gallagher, John Sill (2007).Galaxies in the Universe: An Introduction(2nd ed.). Cambridge University Press. p. 37.ISBN 978-0-521-85593-8.

[Taylor-6] Taylor, John R. (2005).Classical Mechanics.Sausalito, California: University Science Books. p. 29.

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v t e Orthogonal coordinate systems
Two dimensional	Cartesian Polar(Log-polar) Parabolic Bipolar Elliptic
Three dimensional	Cartesian Cylindrical Spherical Parabolic Paraboloidal Oblate spheroidal Prolate spheroidal Ellipsoidal Elliptic cylindrical Toroidal Bispherical Bipolar cylindrical Conical 6-sphere