Cylindrical Coordinates

Cylindrical coordinates is a method of describing location in a three-dimensional coordinate system.

In a cylindrical coordinate system, the location of a three-dimensional point is decribed with the first two dimensions described by polar coordinates and the third dimension described in distance from the plane containing the other two axes. One way to describe cylindrical coordinates is $(r,\theta, z)$, where:

• $r$ is the distance from the origin to the point in the $xy$-plane directly under (or above) the point we are describing
• $\theta$ is the angle between the $x$-axis and that same point in the $xy$-plane
• $z$ is the height of the point

These coordinates are called cylindrical (in contrast to rectangular coordinates or spherical coordinates) because they describe a point on a cylindrical shape (see image above). Problems can become simpler or more difficult when expressed in different coordinate systems and cylindrical coordinates are very helpful when certain kinds of spherical symmetry are present.

Converting to rectangular coordinates involves the same process as converting polar coordinates to cartesian since the first two coordinates in cylindrical coordinates are identical to two-dimensional polar coordinates. To convert from cylindrical coordinates $(r, \theta, z)$ to rectangular coordinates $(a, b, c)$ find $a$, $b$, and $c$ as follows:

• $a=r \cos \theta$
• $b=r \sin \theta$
• $c=z$

To convert from rectangular coordinates $(a, b, c)$ to cylindrical $(r, \theta, z)$, find $r$, $\theta$, and $z$ as follows:

• $r = \sqrt{a^2 + b^2}$
• $\tan \theta = \frac{b}{a}$
• $z=c$

Convert $(3, 60^\circ, 5)$ from cylindrical to rectangular coordinates.

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We know that $x= 3 \cos 60^\circ=\frac{3}{2}$ and $y=3 \sin 60^\circ=\frac{3\sqrt{3}}{2}$, so we will have $\left(\frac32,\frac{3\sqrt{3}}{2},5\right)$.

Spherical coordinates, $(p,\theta,\phi)$, are similar to cylindrical coordinates, but instead of $z$ given in terms of height, spherical coordinates use $\phi$ which is the angle of elevation to the point (or, in some cases, the angle of inclination). Also, $r$ is replaced with $p$, which is the distance between the point being described and the origin. Since there's some differing interpretations of spherical coordinate conventions, you should consult the spherical coordinates page for instructions on conversion.

• Polar Coordinates
• Spherical Coordinates