Archive for February 19th, 2009

Hyperbolic Light

February 19, 2009

A friend of my parents recently sent me a short article entitled “The Shape of Lamp Shade Shadows” by Kenneth E. Horst (The Physics Teacher, Volume 39, March 2001).  In it, the author explains how a friend of his wondered if the shape created when light goes through a lampshade might be hyperbolic:

Disclaimer:  There’s a much better photo in the article itself.  This one makes our living room look much browner than it actually is.  Also much cleaner.

Horst then collected data, analyzed it, and discovered that yes indeed, each curve was in the shape was a hyperbola!  In fact, if $A$ is the vertical distance from the center of the lightbulb to the circular opening on the top or bottom of the lampshade, and $R$ is the radius of the opening, and $D$ is the horizontal distance from the center of the lightbulb to the wall, then the equation of the hyperbola is:

$z=\frac{A}{R}\sqrt{x^2+D^2}$

(with a negative added for the bottom curve).  The top and bottom curves typically come from different hyperbolas, however, because while $D$ is the same in both cases, the top of the lampshade typically has a smaller radius than the bottom; likewise, the bulb is usually closer to the top than the bottom.

In addition to the data evidence that it is a hyperbola, there’s a geometric reason:  the light that leaves the top (and bottom) can be thought of as a cone with vertex in the center of the light bulb, and the wall acts as a vertical cross section:

With this in mind, it might be possible to create the other conic sections by tipping the lampshade (or moving the entire lamp) so that the wall is in different positions relative to the cone of light.  I’m also tempted to build a lampshade that has completely vertical sides with the lightbulb right in the center, so that the top and bottom curves are both part of the same hyperbola.