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

h

) which measures the height of a point above the plane. A point P is given as

(r, \theta, h)

. In terms of the Cartesian coordinate system:

$r$ 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.
$\theta$ is the angle between the positive x-axis and the line OP', measured anti-clockwise.
$h$ is the same as $z$ .

Some mathematicians indeed use

(r, \theta, z)

. 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.

Conversion from cylindrical to Cartesian coordinates

x = r \cos\theta

y = r \sin\theta

z = h

\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}

Conversion from Cartesian to cylindrical coordinates

r = \sqrt{x^2 + y^2}

\theta = \arctan\frac{y}{x}

h = z\,

\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}

Conversion from cylindrical to spherical coordinates

{\rho} =  \sqrt{r^2+h^2}

{\phi} =  \theta \qquad

{\theta'} = \arctan\frac{h}{r} \qquad

\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}

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

Conversion from spherical to cylindrical coordinates

{r} =  \rho \cos \theta

{\theta'} =  \phi

{h} =  \rho \sin \theta

\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}

where φ is azimuth and θ is latitude.