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cylindrical coordinates (Definition)

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

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

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.



"cylindrical coordinates" is owned by Wkbj79.
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See Also: polar coordinates, spherical coordinates
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Cross-references: lying on, point, Cartesian coordinates, azimuthal angle, distance, axis, polar coordinates, coordinates
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This is version 3 of cylindrical coordinates, born on 2007-05-02, modified 2007-05-07.
Object id is 9317, canonical name is CylindricalCoordinates.
Accessed 1115 times total.

Classification:
AMS MSC: 51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)
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