Cylindrical coordinate system

The three coordinates (, , ) of a point are defined as: • The 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 80000-2:2019 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. of the cylindrical coordinates . The red cylinder shows the points with , the blue plane shows the points with , and the yellow half-plane shows the points with . The -axis is vertical and the -axis is highlighted in green. The three surfaces intersect at the point with those coordinates (shown as a black sphere); the Cartesian coordinates of are roughly (1.0, −1.732, 1.0). 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 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{align} x &= \rho \cos \varphi \\ y &= \rho \sin \varphi \\ z &= z \end{align} in one direction, and \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 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 function that returns also an angle in the range = , one may also compute \varphi without computing \rho first \begin{align} \varphi &= \begin{cases} \text{indeterminate} & \text{if } x = 0 \text{ and } y = 0\\ \frac\pi2\frac 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 For other formulas, see the article Polar coordinate system. 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 to or from cylindrical coordinates, depending on whether represents elevation or inclination, by the following: ==Line and volume elements==

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 \mathrm{d}\boldsymbol{r} = \mathrm{d}\rho\,\boldsymbol{\hat{\rho}} + \rho\,\mathrm{d}\varphi\,\boldsymbol{\hat{\varphi}} + \mathrm{d}z\,\boldsymbol{\hat{z}}. The volume element is \mathrm{d}V = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi\,\mathrm{d}z. The surface element in a surface of constant radius (a vertical cylinder) is \mathrm{d}S_\rho = \rho\,\mathrm{d}\varphi\,\mathrm{d}z. The surface element in a surface of constant azimuth (a vertical half-plane) is \mathrm{d}S_\varphi = \mathrm{d}\rho\,\mathrm{d}z. The surface element in a surface of constant height (a horizontal plane) is \mathrm{d}S_z = \rho\,\mathrm{d}\rho\,\mathrm{d}\varphi. The del operator in this system leads to the following expressions for gradient, divergence, curl and Laplacian: \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} ==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 \boldsymbol{r} = \rho\,\boldsymbol{\hat \rho} + z\,\boldsymbol{\hat z}. The velocity of the particle is the time derivative of its position, \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}}, where the term \rho \dot\varphi\hat\varphi comes from the Poisson formula \frac{\mathrm d\hat\rho}{\mathrm dt} = \dot\varphi\hat z\times \hat\rho . Its acceleration is \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}} ==See also==