Cylindrical coordinate system

From Academic Kids

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

A point P is given as <math>(r, \theta, h)<math>. In terms of the Cartesian coordinate system:

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

Some mathematicians indeed use <math>(r, \theta, z)<math>.

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.

Contents

Conversion from cylindrical to Cartesian coordinates

<math>x = r \cos\theta<math>
<math>y = r \sin\theta<math>
<math>z = h<math>

             

<math> \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} <math>

Conversion from Cartesian to cylindrical coordinates

<math>r = \sqrt{x^2 + y^2}<math>
<math>\theta = \arctan\frac{y}{x}<math>
<math>h = z\,<math>

             

<math> \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}

<math>

Conversion from cylindrical to spherical coordinates

<math>{\rho} = \sqrt{r^2+h^2}<math>
<math>{\phi} = \theta \qquad <math>
<math>{\theta'} = \arctan\frac{h}{r} \qquad <math>

             

<math> \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}

<math>

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

Conversion from spherical to cylindrical coordinates

<math>{r} = \rho \cos \theta <math>
<math>{\theta'} = \phi <math>
<math>{h} = \rho \sin \theta <math>

             

<math> \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} <math>

where φ is azimuth and θ is latitude.

See also

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