# Cylindrical coordinate system

The cylindrical coordinate system is a three-dimensional system which essentially extends circular polar coordinates by adding a third coordinate (usually denoted [itex]h[itex]) which measures the height of a point above the plane.

A point P is given as [itex](r, \theta, h)[itex]. In terms of the Cartesian coordinate system:

• [itex]r[itex] is the distance from O to P', the orthogonal projection of the point P onto the XY plane. This is the same as the distance of P to the z-axis.
• [itex]\theta[itex] is the angle between the positive x-axis and the line OP', measured anti-clockwise.
• [itex]h[itex] is the same as [itex]z[itex].

Some mathematicians indeed use [itex](r, \theta, z)[itex].

Cylindrical coordinates are useful in analyzing surfaces that are symmetrical about an axis, with the z-axis chosen as the axis of symmetry. For example, the infinitely long circular cylinder that has the Cartesian equation x2 + y2 = c2 has the very simple equation r = c in cylindrical coordinates. Hence the name "cylindrical" coordinates.

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## Conversion from cylindrical to Cartesian coordinates

 [itex]x = r \cos\theta[itex] [itex]y = r \sin\theta[itex] [itex]z = h[itex] [itex] \begin{vmatrix}dx\\dy\\dz\end{vmatrix} = \begin{vmatrix} \cos\theta&-r\sin\theta&0\\ \sin\theta&r\cos\theta&0\\ 0&0&1 \end{vmatrix} \cdot \begin{vmatrix}dr\\d\theta\\dh\end{vmatrix} [itex]

## Conversion from Cartesian to cylindrical coordinates

 [itex]r = \sqrt{x^2 + y^2}[itex] [itex]\theta = \arctan\frac{y}{x}[itex] [itex]h = z\,[itex] [itex] \begin{vmatrix}dr\\d\theta\\dh\end{vmatrix} = \begin{vmatrix} \frac{x}{\sqrt{x^2+y^2}}&\frac{y}{\sqrt{x^2+y^2}}&0\\ \frac{-y}{x^2+y^2}&\frac{x}{x^2+y^2}&0\\ 0&0&1 \end{vmatrix} \cdot \begin{vmatrix}dx\\dy\\dz\end{vmatrix} [itex]

## Conversion from cylindrical to spherical coordinates

 [itex]{\rho} = \sqrt{r^2+h^2}[itex] [itex]{\phi} = \theta \qquad [itex] [itex]{\theta'} = \arctan\frac{h}{r} \qquad [itex] [itex] \begin{vmatrix}d\rho\\d\phi\\d\theta' \end{vmatrix} = \begin{vmatrix} \frac{r}{\sqrt{r^2+h^2}} & 0 & \frac{h}{\sqrt{r^2+h^2}} \\ 0 & 1 & 0 \\ \frac{-h}{r^2+h^2} & 0 & \frac{r}{r^2+h^2} \end{vmatrix} \cdot \begin{vmatrix}dr\\d\theta\\dh\end{vmatrix} [itex]

where φ is the azimuth and θ' is the latitude.

## Conversion from spherical to cylindrical coordinates

 [itex]{r} = \rho \cos \theta [itex] [itex]{\theta'} = \phi [itex] [itex]{h} = \rho \sin \theta [itex] [itex] \begin{vmatrix}dr\\d\theta'\\dh\end{vmatrix} = \begin{vmatrix} \cos \theta & 0 & - \rho \sin \theta \\ 0 & 1 & 0 \\ \sin \theta & 0 & \rho \cos \theta \end{vmatrix} \cdot \begin{vmatrix}d\rho\\d\phi\\d\theta\end{vmatrix} [itex]

where φ is azimuth and θ is latitude.

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