Cylindrical coordinate system

A cylindrical coordinate system is a three-dimensional coordinate system that 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.

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. 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 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,^[1] or axial position.^[2]

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 disks in 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]

The radial distance $ρ$ is the from the $z$ -axis to the point $P$ .

Euclidean distance

The azimuth $φ$ 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.

The axial coordinate or height $z$ is the signed distance from the chosen plane to the point $P$ .

Cylindrical harmonics[edit]

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

List of canonical coordinate transformations

Vector fields in cylindrical and spherical coordinates

Del in cylindrical and spherical coordinates

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Boston

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. 978-0-387-18430-2.