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A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions around a main axis (a chosen directed line) and an auxiliary axis (a reference ray). The three cylindrical coordinates are: the point perpendicular distance Template:Math from the main axis; the point signed distance z along the main axis from a chosen origin; and the plane angle Template:Math of the point projection on a reference plane (passing through the origin and perpendicular to the main axis)
The main axis is variously called the cylindrical or longitudinal axis. The auxiliary axis is called the polar axis, which 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,<ref>Template:Cite journal</ref> or axial position.<ref>Template:Cite journal</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 disks in astronomy, and so on.
They are sometimes called cylindrical polar coordinates<ref>Template:Cite book</ref> or polar cylindrical coordinates,<ref>Template:Cite book</ref> and are sometimes used to specify the position of stars in a galaxy (galactocentric cylindrical polar coordinates).<ref>Template:Cite book</ref>
DefinitionEdit
The three coordinates (Template:Mvar, Template:Mvar, Template:Mvar) of a point Template:Mvar are defined as:
- The radial distance Template:Mvar is the Euclidean distance from the Template:Mvar-axis to the point Template:Mvar.
- The azimuth Template:Mvar is the angle between the reference direction on the chosen plane and the line from the origin to the projection of Template:Mvar on the plane.
- The axial coordinate or height Template:Mvar is the signed distance from the chosen plane to the point Template:Mvar.
Unique cylindrical coordinatesEdit
As in polar coordinates, the same point with cylindrical coordinates Template:Math has infinitely many equivalent coordinates, namely Template:Math and Template:Math where Template:Mvar is any integer. Moreover, if the radius Template:Mvar 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 (Template:Math) and the azimuth Template:Mvar to lie in a specific interval spanning 360°, such as Template:Math or Template:Math.
ConventionsEdit
The notation for cylindrical coordinates is not uniform. The ISO standard 31-11 recommends Template:Math, where Template:Mvar is the radial coordinate, Template:Mvar the azimuth, and Template:Mvar the height. However, the radius is also often denoted Template:Mvar or Template:Mvar, the azimuth by Template:Mvar or Template:Mvar, and the third coordinate by Template:Mvar or (if the cylindrical axis is considered horizontal) Template:Mvar, or any context-specific letter.
In concrete situations, and in many mathematical illustrations, a positive angular coordinate is measured counterclockwise as seen from any point with positive height.
Coordinate system conversionsEdit
The cylindrical coordinate system is one of many three-dimensional coordinate systems. The following formulae may be used to convert between them.
Cartesian coordinatesEdit
For the conversion between cylindrical and Cartesian coordinates, it is convenient to assume that the reference plane of the former is the Cartesian Template:Mvar-plane (with equation Template:Math), and the cylindrical axis is the Cartesian Template:Mvar-axis. Then the Template:Mvar-coordinate is the same in both systems, and the correspondence between cylindrical Template:Math and Cartesian Template:Math are the same as for polar coordinates, namely <math display="block"> \begin{align} x &= \rho \cos \varphi \\ y &= \rho \sin \varphi \\ z &= z \end{align} </math> in one direction, and <math display="block">\begin{align} \rho &= \sqrt{x^2+y^2} \\ \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\ \arcsin\left(\frac{y}{\rho}\right) & \text{if } x \geq 0 \\ -\arcsin\left(\frac{y}{\rho}\right) + \pi & \mbox{if } x < 0 \text{ and } y \ge 0\\ -\arcsin\left(\frac{y}{\rho}\right) - \pi & \mbox{if } x < 0 \text{ and } y < 0 \end{cases} \end{align}</math> in the other. The arcsine function is the inverse of the sine function, and is assumed to return an angle in the range Template:Math = Template:Math. These formulas yield an azimuth Template:Mvar in the range Template:Math.
By using the arctangent function that returns also an angle in the range Template:Math = Template:Math, one may also compute <math>\varphi</math> without computing <math>\rho</math> first <math display="block">\begin{align} \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\ \frac\pi2\frac y{|y|} & \text{if } x = 0 \text{ and } y \ne 0\\ \arctan\left(\frac{y}{x}\right) & \mbox{if } x > 0 \\ \arctan\left(\frac{y}{x}\right)+\pi & \mbox{if } x < 0 \text{ and } y \ge 0\\ \arctan\left(\frac{y}{x}\right)-\pi & \mbox{if } x < 0 \text{ and } y < 0 \end{cases} \end{align}</math> For other formulas, see the article Polar coordinate system.
Many modern programming languages provide a function that will compute the correct azimuth Template:Mvar, in the range Template:Math, given x and y, without the need to perform a case analysis as above. For example, this function is called by Template:Mono in the C programming language, and Template:Mono in Common Lisp.
Spherical coordinatesEdit
Spherical coordinates (radius Template:Mvar, elevation or inclination Template:Mvar, azimuth Template:Mvar), may be converted to or from cylindrical coordinates, depending on whether Template:Mvar represents elevation or inclination, by the following:
| Conversion to: | Coordinate | Template:Mvar is elevation | Template:Mvar is inclination |
|---|---|---|---|
| Cylindrical | Template:Mvar = | Template:Math | Template:Math |
| Template:Mvar = | Template:Mvar | ||
| Template:Mvar = | Template:Math | Template:Math | |
| Spherical | Template:Mvar = | <math display="inline">\sqrt{\rho^2+z^2}</math> | |
| Template:Mvar = | <math display="inline">\arctan\left(\frac{z}{\rho}\right)</math> | <math display="inline">\arctan\left(\frac{\rho}{z}\right)</math> | |
| Template:Mvar = | Template:Mvar | ||
Line and volume elementsEdit
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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 <math display="block">\mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}.</math>
The volume element is <math display="block">\mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>
The surface element in a surface of constant radius Template:Mvar (a vertical cylinder) is <math display="block">\mathrm{d}S_\rho = \rho\,\mathrm{d}\varphi\,\mathrm{d}z.</math>
The surface element in a surface of constant azimuth Template:Mvar (a vertical half-plane) is <math display="block">\mathrm{d}S_\varphi = \mathrm{d}\rho\,\mathrm{d}z.</math>
The surface element in a surface of constant height Template:Mvar (a horizontal plane) is <math display="block">\mathrm{d}S_z = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi.</math>
The del operator in this system leads to the following expressions for gradient, divergence, curl and Laplacian: <math display="block">\begin{align}
\nabla f &= \frac{\partial f}{\partial \rho}\boldsymbol{\hat{\rho}} + \frac{1}{\rho}\frac{\partial f}{\partial \varphi}\boldsymbol{\hat{\varphi}} + \frac{\partial f}{\partial z}\boldsymbol{\hat{z}} \\[8px]
\nabla \cdot \boldsymbol{A} &= \frac{1}{\rho}\frac{\partial}{\partial \rho}\left(\rho A_\rho\right) + \frac{1}{\rho} \frac{\partial A_\varphi}{\partial \varphi} + \frac{\partial A_z}{\partial z} \\[8px]
\nabla \times \boldsymbol{A} &= \left(\frac{1}{\rho}\frac{\partial A_z}{\partial \varphi} - \frac{\partial A_\varphi}{\partial z}\right)\boldsymbol{\hat{\rho}} +
\left(\frac{\partial A_\rho}{\partial z} - \frac{\partial A_z}{\partial \rho}\right)\boldsymbol{\hat{\varphi}} +
\frac{1}{\rho}\left(\frac{\partial}{\partial \rho}\left(\rho A_\varphi\right) - \frac{\partial A_\rho}{\partial \varphi}\right) \boldsymbol{\hat{z}} \\[8px]
\nabla^2 f &= \frac{1}{\rho} \frac{\partial}{\partial \rho} \left(\rho \frac{\partial f}{\partial \rho}\right) +
\frac{1}{\rho^2} \frac{\partial^2 f}{\partial \varphi^2} + \frac{\partial^2 f}{\partial z^2}
\end{align}</math>
Cylindrical harmonicsEdit
The solutions to the Laplace equation in a system with cylindrical symmetry are called cylindrical harmonics.
KinematicsEdit
In a cylindrical coordinate system, the position of a particle can be written as<ref name="Taylor">Template:Cite book</ref> <math display="block">\boldsymbol{r} = \rho\,\boldsymbol{\hat \rho} + z\,\boldsymbol{\hat z}.</math> The velocity of the particle is the time derivative of its position, <math display="block">\boldsymbol{v} = \frac{\mathrm{d}\boldsymbol{r}}{\mathrm{d}t} = \dot{\rho}\,\boldsymbol{\hat \rho} + \rho\,\dot\varphi\,\hat{\boldsymbol{\varphi}} + \dot{z}\,\hat{\boldsymbol{z}},</math> where the term <math>\rho \dot\varphi\hat\varphi</math> comes from the Poisson formula <math>\frac{\mathrm d\hat\rho}{\mathrm dt} = \dot\varphi\hat z\times \hat\rho </math>. Its acceleration is<ref name="Taylor"/> <math display="block"> \boldsymbol{a} = \frac{\mathrm{d}\boldsymbol{v}}{\mathrm{d}t} = \left( \ddot{\rho} - \rho\,\dot\varphi^2 \right)\boldsymbol{\hat \rho} + \left( 2\dot{\rho}\,\dot\varphi + \rho\,\ddot\varphi \right) \hat{\boldsymbol\varphi } + \ddot{z}\,\hat{\boldsymbol{z}} </math>
See alsoEdit
- List of canonical coordinate transformations
- Vector fields in cylindrical and spherical coordinates
- Del in cylindrical and spherical coordinates
ReferencesEdit
Further readingEdit
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Cite book
External linksEdit
- Template:Springer
- MathWorld description of cylindrical coordinates
- Cylindrical Coordinates Animations illustrating cylindrical coordinates by Frank Wattenberg
Template:Orthogonal coordinate systems
de:Polarkoordinaten#Zylinderkoordinaten ro:Coordonate polare#Coordonate cilindrice fi:Koordinaatisto#Sylinterikoordinaatisto