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 
(with a negative added for the bottom curve). The top and bottom curves typically come from different hyperbolas, however, because while
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.
Thanks to Ted Foster for sending me this article!
360 
February 19, 2009 at 12:45 pm |
You should also black out the shade so you don’t get the side pieces
February 19, 2009 at 3:23 pm |
This assumes that the light is a point source which is clearly not true. If you assume it’s a radiating sphere, the shape changes.
February 19, 2009 at 4:52 pm |
Sure, it changes… to a fuzzy hyperbola.
February 19, 2009 at 6:16 pm |
I knew this before, but that diagram at the end really stood out. It’s amazing. Sometimes when I see things like this, I wish I taught precalculus. That ah ha moment after students have played around and talked and sketched and everything — and then they decide it is a hyperbola. . . and then the teacher throws up that one image? It all comes together.
Sam
February 20, 2009 at 9:41 am |
Hi great post!!!!…I really enjoyed
February 20, 2009 at 10:03 am |
Thanks!
Dave: good idea about blacking out the lampshade. There’s a little extra light in this case because that shade has a hole in the back (a reminder to turn off the lights whenever leaving the house, after one of our cats knocked the lamp over and it quickly burned the shade).
Sam, to give credit, the underlying drawing is public domain from Scott Foresman (here).
February 20, 2009 at 10:08 am |
Hi…I wonder if you can share the original article….thanks in advance!!!
February 20, 2009 at 10:23 am |
Tibu, I looked for a copy on the web, but couldn’t find one. The closest I found was here, but you have to be logged in to read it. (I’d expect that most libraries could get it through interlibrary loan, however.)
February 27, 2009 at 6:48 am |
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March 5, 2012 at 4:55 am |
Could you provide an explanation on how the formula z=A/R sqrt(x^2+D^2) was obtained?
Thank you very much in advance.