Recently, Ξ over at the 360 blog wrote about hyperbolic light (which was inspired by the article, “The Shape of Lamp Shade Shadows” by Kenneth E. Horst, *The Physics Teacher*, Volume 39, March 2001). They were looking at the pattern of light on the wall emitted by a desk lamp with a cylindrical lampshade. They observed that the shape of the light was a hyperbola. The reason for this, as Ξ points out, is that the bulb and lampshade form a cone of light and the wall is a plane slicing that cone. As we see in the third picture below, such a plane cuts the cone in a hyperbola.

It stands to reason that we can get the other conic sections—the parabola (left), circle (center, bottom) and ellipse (center, top)—by tilting the lamp.

As it turned out I was teaching about conic sections last week (they arise as traces of quadric surfaces), so I decided I had to show this to my class. I walk to work and didn’t want to carry a lamp in to school, so I decided to make a portable version.

My first thought was to use a flashlight. But flashlights are designed to focus light to a single point as much as possible. They don’t make a nice cone. Then I decided to make a “lampshade” (shown below) for my flashlight out of a stiff piece of paper (I used a file folder). Ideally there would be a single point source of light at the bottom of the shade, but this won’t be the case because of the parabolic reflector in the flashlight. After a little experimentation I found that the longer the “shade,” the better the cone of light. Also, to make a circular cone, the shade needs a good circular opening at the other end (not squashed or kinked)—it took a little fussing to get things the way I wanted them.

In class class I drew the blinds, turned off the lights, and shined my light on the projector screen. It worked pretty well. I could make a good circle (point the flashlight directly at the wall), an ellipse (tip the light up a little), a parabola (tip it farther), and half a hyperbola (farther still)!

The students thought it was pretty cool (I have a good bunch this semester!).

[Conic section image by Pbroks13, published under the Creative Commons Attribution 3.0 Unported License.]

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*Related*

A conic section cannot be a perfect ellipse which is symmetrical East-West as well as North-South. A perfect ellipse is a cylindric section.

It is surprising, but will be symmetrical (http://mathworld.wolfram.com/ConicSection.html).

The locations for the foci are identifiable on the cylindric section but not on the conic section.