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A cylindrical coordinate system is a three-dimensional
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
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 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 the ...
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 Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
about the longitudinal axis, such as water flow in a straight pipe with round cross-section, heat distribution in a metal
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...
,
electromagnetic fields An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical co ...
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 moving pa ...
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 In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
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 In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
() 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 ISO is the most common abbreviation for the International Organization for Standardization. ISO or Iso may also refer to: Business and finance * Iso (supermarket), a chain of Danish supermarkets incorporated into the SuperBest chain in 2007 * 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 ...
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 \begin x &= \rho \cos \varphi \\ y &= \rho \sin \varphi \\ z &= z \end in one direction, and \begin \rho &= \sqrt \\ \varphi &= \begin \text & \text x = 0 \text y = 0\\ \arcsin\left(\frac\right) & \text x \geq 0 \\ -\arcsin\left(\frac\right) + \pi & \mbox x < 0 \text y \ge 0\\ -\arcsin\left(\frac\right) + \pi & \mbox x < 0 \text y < 0 \end \end in the other. The
arcsine 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). Spec ...
function is the inverse of the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
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). Spec ...
function that returns also an angle in the range = , one may also compute \varphi without computing \rho first \begin \varphi &= \begin \text & \text x = 0 \text y = 0\\ \frac\pi2\frac y & \text x = 0 \text y \ne 0\\ \arctan\left(\frac\right) & \mbox x > 0 \\ \arctan\left(\frac\right)+\pi & \mbox x < 0 \text y \ge 0\\ \arctan\left(\frac\right)-\pi & \mbox x < 0 \text y < 0 \end \end 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 the or ...
. 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 Common Lisp (CL) is a dialect of the Lisp programming language, published in ANSI standard document ''ANSI INCITS 226-1994 (S20018)'' (formerly ''X3.226-1994 (R1999)''). The Common Lisp HyperSpec, a hyperlinked HTML version, has been derived fro ...
.


Spherical coordinates

Spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
(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 In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, or . Integrals of a function of two variables over a region in \mathbb^2 (the real-number ...
for details of volume integration in cylindrical coordinates, and
Del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinates, curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11#Coordinate systems, ISO 31- ...
for
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
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 In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc l ...
is :\mathrm\boldsymbol = \mathrm\rho\,\boldsymbol + \rho\,\mathrm\varphi\,\boldsymbol + \mathrmz\,\boldsymbol. The
volume element In mathematics, a volume element provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form :dV ...
is :\mathrmV = \rho\,\mathrm\rho\,\mathrm\varphi\,\mathrmz. The surface element in a surface of constant radius (a vertical cylinder) is :\mathrmS_\rho = \rho\,\mathrm\varphi\,\mathrmz. The surface element in a surface of constant azimuth (a vertical half-plane) is :\mathrmS_\varphi = \mathrm\rho\,\mathrmz. The surface element in a surface of constant height (a horizontal plane) is :\mathrmS_z = \rho\,\mathrm\rho\,\mathrm\varphi. The
del Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes ...
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 gradi ...
,
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...
,
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...
and
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
: :\begin \nabla f &= \frac\boldsymbol + \frac\frac\boldsymbol + \frac\boldsymbol \\ px \nabla \cdot \boldsymbol &= \frac\frac\left(\rho A_\rho\right) + \frac \frac + \frac \\ px \nabla \times \boldsymbol &= \left(\frac\frac - \frac\right)\boldsymbol + \left(\frac - \frac\right)\boldsymbol + \frac\left(\frac\left(\rho A_\varphi\right) - \frac\right) \boldsymbol \\ px \nabla^2 f &= \frac \frac \left(\rho \frac\right) + \frac \frac + \frac \end


Cylindrical harmonics

The solutions to the
Laplace equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
in a system with cylindrical symmetry are called
cylindrical harmonics In mathematics, the cylindrical harmonics are a set of linearly independent functions that are solutions to Laplace's differential equation, \nabla^2 V = 0, expressed in cylindrical coordinates, ''ρ'' (radial coordinate), ''φ'' (polar angle), an ...
.


Kinematics

In a cylindrical coordinate system, the position of a particle can be written as \boldsymbol = \rho\,\boldsymbol + \varphi\,\hat + z\,\boldsymbol. The velocity of the particle is the time derivative of its position, \boldsymbol = \frac = \dot\,\boldsymbol + \rho\,\dot\varphi\,\hat + \dot\,\hat, and its acceleration is \boldsymbol = \frac = \left( \ddot - \rho\,\dot\varphi^2 \right)\boldsymbol + \left( 2\dot\,\dot\varphi + \rho\,\ddot\varphi \right) \hat + \ddot\,\hat


See also

*
List of canonical coordinate transformations This is a list of some of the most commonly used coordinate transformations. 2-dimensional Let (''x'', ''y'') be the standard Cartesian coordinates, and (''r'', ''θ'') the standard polar coordinates. To Cartesian coordinates From polar coordi ...
*
Vector fields in cylindrical and spherical coordinates Note: This page uses common physics notation for spherical coordinates, in which \theta is the angle between the ''z'' axis and the radius vector connecting the origin to the point in question, while \phi is the angle between the projection of t ...
*
Del in cylindrical and spherical coordinates This is a list of some vector calculus formulae for working with common curvilinear coordinates, curvilinear coordinate systems. Notes * This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11#Coordinate systems, ISO 31- ...


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 de:Polarkoordinaten#Zylinderkoordinaten ro:Coordonate polare#Coordonate cilindrice fi:Koordinaatisto#Sylinterikoordinaatisto