Hyperbolic Light

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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:

lampshadeDisclaimer:  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:

hyperbola_with-lampshade

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!

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11 Responses to “Hyperbolic Light”

  1. David Petersen Says:

    You should also black out the shade so you don’t get the side pieces

  2. Hang Says:

    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.

  3. Andrew Says:

    Sure, it changes… to a fuzzy hyperbola.

  4. samjshah Says:

    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

  5. tibu Says:

    Hi great post!!!!…I really enjoyed

  6. Ξ Says:

    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).

  7. tibu Says:

    Hi…I wonder if you can share the original article….thanks in advance!!!

  8. Ξ Says:

    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.)

  9. 50th Carnival of Mathematics — The Endeavour Says:

    [...] from 360 presents Hyperbolic Light, explaining why a lamp casts a hyperbolic pattern of light on a [...]

  10. Lámpara hiperbólica « Apuntes Matemáticos Says:

    [...] Hyperbolic Light – entrada del 19/02/2009 del  blog [...]

  11. Hm Says:

    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.

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