# cylindrical coordinates

Cylindrical coordinates are a system of coordinates for $\mathbb{R}^{3}$. Two of the coordinates correspond to the polar coordinates of $\mathbb{R}^{2}$, and the third coordinate corresponds with the $z$ axis. Thus, the coordinates are given by

 $\left(\begin{array}[]{c}x\\ y\\ z\end{array}\right)=\left(\begin{array}[]{c}r\cos\theta\\ r\sin\theta\\ z\end{array}\right),$

where $r$ is the distance from $(0,0,0)$ to $(x,y,0)$ and $\theta$ is the azimuthal angle defined for $\theta\in[0,2\pi)$.

Just as with polar coordinates, one can convert from Cartesian coordinates to cylindrical coordinates for any point not lying on the $z$ axis via

 $\displaystyle r(x,y)$ $\displaystyle=$ $\displaystyle\sqrt{x^{2}+y^{2}},$ $\displaystyle\theta(x,y)$ $\displaystyle=$ $\displaystyle\arctan(x,y),$

where $\arctan$ is defined here (http://planetmath.org/OperatornamearcTanWithTwoArguments).

Title cylindrical coordinates CylindricalCoordinates 2013-03-22 17:01:54 2013-03-22 17:01:54 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Definition msc 51M05 PolarCoordinates SphericalCoordinates