## ::Cylindrical coordinate system

### ::concepts

Partial::varphi    Mathrm::polar    ''z''::first    Plane::point    System::azimuth    Title::distance A cylindrical coordinate system with origin O, polar axis A, and longitudinal axis L. The dot is the point with radial distance ρ = 4, angular coordinate φ = 130°, and height z = 4.

A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the axis. The latter distance is given as a positive or negative number depending on which side of the reference plane faces the point.

The origin of 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 the cylindrical or longitudinal axis, to differentiate it from the polar axis, which is the ray that lies in the reference plane, starting at the origin and pointing in the reference direction.

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 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>{{#invoke:Citation/CS1|citation |CitationClass=journal }}</ref> or axial position.<ref>{{#invoke:Citation/CS1|citation |CitationClass=journal }} "[...]where r, θ, and z are cylindrical coordinates [...] as a function of axial position[...]"</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, electromagnetic fields produced by an electric current in a long, straight wire, accretion discs in astronomy, and so on.

It is sometimes called "cylindrical polar coordinate"<ref>J. E. Szymanski, Basic mathematics for electronic engineers: models and applications, Volume 16 of Tutorial guides in electronic engineering, Publisher Taylor & Francis, 1989, ISBN 0-278-00068-1, ISBN 978-0-278-00068-1 (page 170)</ref> and "polar cylindrical coordinate",<ref>Robert H. Nunn, Intermediate fluid mechanics, Publisher Taylor & Francis, 1989, ISBN 0-89116-647-5, ISBN 978-0-89116-647-4, 343 pages (page 3)</ref> and is sometimes used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinate").<ref>Linda Siobhan Sparke, John Sill Gallagher, Galaxies in the universe: an introduction, Edition 2, Publisher Cambridge University Press, 2007, ISBN 0-521-85593-4, ISBN 978-0-521-85593-8, 431 pages (page 37)</ref>

Cylindrical coordinate system sections