Cylindrical coordinate system
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0e/Coord_system_CY_1.svg/240px-Coord_system_CY_1.svg.png)
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
[edit]The three coordinates (ρ,φ,z) of a pointPare defined as:
- Theradial distanceρis theEuclidean distancefrom thez-axis to the pointP.
- Theazimuthφis the angle between the reference direction on the chosen plane and the line from the origin to the projection ofPon the plane.
- Theaxial coordinateorheightzis the signed distance from the chosen plane to the pointP.
Unique cylindrical coordinates
[edit]As in polar coordinates, the same point with cylindrical coordinates(ρ,φ,z)has infinitely many equivalent coordinates, namely(ρ,φ±n×360°,z)and(−ρ,φ± (2n+ 1)×180°,z),wherenis any integer. Moreover, if the radiusρis 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≥ 0) and the azimuthφto lie in a specificintervalspanning 360°, such as[−180°,+180°]or[0,360°].
Conventions
[edit]The notation for cylindrical coordinates is not uniform. TheISOstandard31-11recommends(ρ,φ,z),whereρis the radial coordinate,φthe azimuth, andzthe height. However, the radius is also often denotedrors,the azimuth byθort,and the third coordinate byhor (if the cylindrical axis is considered horizontal)x,or any context-specific letter.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Cylindrical_coordinate_surfaces.png/220px-Cylindrical_coordinate_surfaces.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Cylindrical_coordinate_surfaces.gif/220px-Cylindrical_coordinate_surfaces.gif)
In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measuredcounterclockwiseas seen from any point with positive height.
Coordinate system conversions
[edit]The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be used to convert between them.
Cartesian coordinates
[edit]For the conversion between cylindrical and Cartesian coordinates, it is convenient to assume that the reference plane of the former is the Cartesianxy-plane (with equationz= 0), and the cylindrical axis is the Cartesianz-axis. Then thez-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 in one direction, and in the other. Thearcsinefunction is the inverse of thesinefunction, and is assumed to return an angle in the range[−π/2,+π/2]=[−90°, +90°].These formulas yield an azimuthφin the range[−90°, +270°].
By using thearctangentfunction that returns also an angle in the range[−π/2,+π/2]=[−90°, +90°],one may also computewithout computingfirst For other formulas, see the articlePolar coordinate system.
Many modern programming languages provide a function that will compute the correct azimuthφ,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
[edit]Spherical coordinates(radiusr,elevation or inclinationθ,azimuthφ), may be converted to or from cylindrical coordinates, depending on whetherθrepresents elevation or inclination, by the following:
Conversion to: | Coordinate | θis elevation | θis inclination |
---|---|---|---|
Cylindrical | ρ= | rcosθ | rsinθ |
φ= | φ | ||
z= | rsinθ | rcosθ | |
Spherical | r= | ||
θ= | |||
φ= | φ |
Line and volume elements
[edit]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
Thevolume elementis
Thesurface elementin a surface of constant radiusρ(a vertical cylinder) is
The surface element in a surface of constant azimuthφ(a vertical half-plane) is
The surface element in a surface of constant heightz(a horizontal plane) is
Thedeloperator in this system leads to the following expressions forgradient,divergence,curlandLaplacian:
Cylindrical harmonics
[edit]The solutions to theLaplace equationin a system with cylindrical symmetry are calledcylindrical harmonics.
Kinematics
[edit]In a cylindrical coordinate system, the position of a particle can be written as[6] The velocity of the particle is the time derivative of its position, where the termcomes from the Poisson formula.Its acceleration is[6]
See also
[edit]- List of canonical coordinate transformations
- Vector fields in cylindrical and spherical coordinates
- Del in cylindrical and spherical coordinates
References
[edit]- ^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.ISSN1089-7674.Archived fromthe originalon 14 April 2013.Retrieved9 February2013.
...in cylindrical coordinates(r,θ,z)... andZ=vbztis 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.S2CID54814721.
...wherer,θ,andzare 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.ISBN978-0-278-00068-1.
- ^Nunn, Robert H. (1989).Intermediate Fluid Mechanics.Taylor & Francis. p. 3.ISBN978-0-89116-647-4.
- ^Sparke, Linda Siobhan;Gallagher, John Sill (2007).Galaxies in the Universe: An Introduction(2nd ed.). Cambridge University Press. p. 37.ISBN978-0-521-85593-8.
- ^abTaylor, John R. (2005).Classical Mechanics.Sausalito, California: University Science Books. p. 29.
Further reading
[edit]- Morse, Philip M.;Feshbach, Herman(1953).Methods of Theoretical Physics, Part I.New York City:McGraw-Hill.pp. 656–657.ISBN0-07-043316-X.LCCN52011515.
- Margenau, Henry;Murphy, George M. (1956).The Mathematics of Physics and Chemistry.New York City: D. van Nostrand. p.178.ISBN9780882754239.LCCN55010911.OCLC3017486.
- Korn, Granino A.;Korn, Theresa M.(1961).Mathematical Handbook for Scientists and Engineers.New York City: McGraw-Hill. pp.174–175.LCCN59014456.ASIN B0000CKZX7.
- Sauer, Robert; Szabó, István (1967).Mathematische Hilfsmittel des Ingenieurs.New York City:Springer-Verlag.p. 95.LCCN67025285.
- Zwillinger, Daniel (1992).Handbook of Integration.Boston:Jones and Bartlett Publishers.p. 113.ISBN0-86720-293-9.OCLC25710023.
- Moon, P.; Spencer, D. E. (1988). "Circular-Cylinder Coordinates (r, ψ, z)".Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions(corrected 2nd ed.). New York City: Springer-Verlag. pp. 12–17, Table 1.02.ISBN978-0-387-18430-2.
External links
[edit]- "Cylinder coordinates",Encyclopedia of Mathematics,EMS Press,2001 [1994]
- MathWorld description of cylindrical coordinates
- Cylindrical CoordinatesAnimations illustrating cylindrical coordinates by Frank Wattenberg