Conversions Between Cartesian, Cylindrical and Spherical Coordinates

Posted by Jeff Schuler on Sep 20, 2009 in Uncategorized |

The 3 common methods of describing a point in a three dimensional coordinate system are Cartesian, Cylindrical and Spherical. The most simple is Cartesian but certain teachers find it necessary to use the others. There are a few simple conversions between them but first it is necessary to know their notation.

Cartesian: (x,y,z)

Cylindrical: (\rho,\phi,z)

Spherical: (r,\theta,\phi)

In most cases you will only need to work from Cartesian to Cylindrical or Spherical OR back, so I will only supply those equations. If you need to work between Cylindrical and Spherical, it would be one more simple step working from one of those, to Cartesian, then on to the other.

Cartesian \rightarrow Cylindrical:

x = \rho \cdot cos (\phi)
y = \rho \cdot sin (\phi)
z = z

Cartesian \leftarrow Cylindrical:

\rho = \sqrt{x^2 + y^2}
\phi = tan^{-1} (\frac{y}{x})
z = z

Cartesian \rightarrow Spherical:

x = r \cdot sin(\theta) \cdot cos(\phi)
y = r \cdot sin(\theta) \cdot sin(\phi)
z = r \cdot cos(\theta)

Cartesian \leftarrow Spherical:

r = \sqrt{x^2 + y^2 + z^2}
\theta = tan^{-1}(\frac{\sqrt{x^2 + y^2}}{z})
\phi = tan^{-1}(\frac{y}{x})

Plug the values from any given points into the correct equation to convert to a different type of coordinate system.

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