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
Ray may refer to:
Fish
* Ray (fish), any cartilaginous fish of the superorder Batoidea
* Ray (fish fin anatomy), a bony or horny spine on a fin
Science and mathematics
* Ray (geometry), half of a line proceeding from an initial point
* Ray (g ...
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'', or ''axial position''.
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
An electric current is a stream of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is measured as the net rate of flow of electric charge through a surface or into a control volume. The movin ...
in a long, straight wire,
accretion disks in astronomy, and so on.
They are sometimes called "cylindrical polar coordinates" and "polar cylindrical coordinates", and are sometimes used to specify the position of stars in a galaxy ("galactocentric cylindrical polar coordinates").
Definition
The three coordinates (, , ) of a point are defined as:
* The ''axial distance'' or ''radial distance'' is the
Euclidean distance from the -axis to the point .
* The ''azimuth'' is the angle between the reference direction on the chosen plane and the line from the origin to the projection of on the plane.
* The ''axial coordinate'' or ''height'' is the signed distance from the chosen plane to the point .
Unique cylindrical coordinates
As in polar coordinates, the same point with cylindrical coordinates has infinitely many equivalent coordinates, namely and where is any integer. Moreover, if the radius 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 be
non-negative () and the azimuth to lie in a specific
interval spanning 360°, such as or .
Conventions
The notation for cylindrical coordinates is not uniform. The
ISO standard
31-11 recommends , where is the radial coordinate, the azimuth, and the height. However, the radius is also often denoted or , the azimuth by or , and the third coordinate by or (if the cylindrical axis is considered horizontal) , or any context-specific letter.

In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured
counterclockwise
Two-dimensional rotation can occur in two possible directions. Clockwise motion (abbreviated CW) proceeds in the same direction as a clock's hands: from the top to the right, then down and then to the left, and back up to the top. The opposite s ...
as 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 -plane (with equation ), and the cylindrical axis is the Cartesian -axis. Then the -coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely
in one direction, and
in the other. The
arcsine function is the inverse of the
sine function, and is assumed to return an angle in the range = . These formulas yield an azimuth in the range .
By using the
arctangent
In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted domains). S ...
function that returns also an angle in the range = , one may also compute
without computing
first
For other formulas, see the article
Polar coordinate system
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
.
Many modern programming languages provide a function that will compute the correct azimuth , in the range , given ''x'' and ''y'', without the need to perform a case analysis as above. For example, this function is called by in the
C programming language, and in
Common Lisp.
Spherical coordinates
Spherical coordinates (radius , elevation or inclination , azimuth ), may be converted into cylindrical coordinates by:
Cylindrical coordinates may be converted into spherical coordinates by:
Line and volume elements
:''See
multiple integral for details of volume integration in cylindrical coordinates, and
Del in cylindrical and spherical coordinates for
vector calculus formulae.''
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.
The
line element is
:
The
volume element is
:
The
surface element in a surface of constant radius (a vertical cylinder) is
:
The surface element in a surface of constant azimuth (a vertical half-plane) is
:
The surface element in a surface of constant height (a horizontal plane) is
:
The
del operator in this system leads to the following expressions for
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
,
divergence,
curl and
Laplacian:
:
Cylindrical harmonics
The solutions to the
Laplace equation in a system with cylindrical symmetry are called
cylindrical harmonics.
Kinematics
In a cylindrical coordinate system, the position of a particle can be written as
The velocity of the particle is the time derivative of its position,
and its acceleration is
See also
*
List of canonical coordinate transformations
*
Vector fields in cylindrical and spherical coordinates
*
Del in cylindrical and spherical coordinates
References
Further reading
*
*
*
*
*
*
External links
*
MathWorld description of cylindrical coordinates Animations illustrating cylindrical coordinates by Frank Wattenberg
{{Orthogonal coordinate systems
Three-dimensional coordinate systems
Orthogonal coordinate systems
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