Last time we used wild properties of ellipses to build some really easy—and some really devilish—golf courses. Specifically, I claimed that every ellipse has two magical points *F _{1}* and

*F*(called

_{2}**foci**) such that a ray from

*F*always bounces off the ellipse and lands precisely at

_{1}*F*, and furthermore, this path always has the same length. Why does this happen? And how do we find these foci?

_{2}Let’s focus(!) on the last question first. Recall that an ellipse is a stretched circle. In other words, an ellipse is what forms when you slice a tall, circular tube (cylinder) along a slant:

Take a sphere that snugly fits inside the tube, and drop it down until it touches the ellipse-slice at a single point *F _{1}*. Do the same with a sphere underneath, touching the slice at

*F*. These points turn out to be the foci of the ellipse. Let’s see why.

_{2}We can use this tubular setup to answer one mystery from earlier: for any point *X* on the ellipse, the sum of distances of *XF _{1}* and

*XF*is always the same! The proof lies in the following animation. Segments

_{2}*XF*and

_{1}*XA*have the same length because they’re both tangent to the upper sphere from

*X*, and similarly,

*XF*. So the sum

_{2}=XB*XF*is just the length of segment

_{1}+XF_{2}*AB*, the height between the spheres’ equators.

This has two neat consequences. First, it provides an elementary method for drawing ellipses (in real life!): all you need are two push pins and a loop of string, as illustrated below. The string ensures that the sum *XF _{1}+XF_{2}* stays fixed while you trace the pen around, as long as you’re careful to keep the string taut throughout.

Second, what happens if we slice a *cone* instead of a cylinder? Perhaps surprisingly, we still get an ellipse! Indeed, as above, we can create a sphere on either side of the slice that snugly fits against the slice and the walls of the cone (the so-called **Dandelin spheres**), and exactly the same proof shows that *XF _{1}+XF_{2}* stays constant as

*X*moves around the edge.

But wait, there’s still an unanswered question! We’ve seen that the path *F _{1}X+XF_{2}* has a fixed length, but why does light bounce off an ellipse along such a path? This is what we really cared about for mini-golf! Come back next time for the answer, and in the meantime, have a great 2 weeks.

Michael,

I’m glad, and thank you for your very nice comments in the last few weeks! =D

Josh,

Thank you too! I think at least part of the reason is that, in high school, conics are often taught just when analytical/coordinate geometry is being introduced. This reinforces valuable algebraic techniques, but at the expense of some pretty geometry.

I LOVE this!

Zach — this is WONDERFUL. It’s hard to feel jaded about mathematical beauty when you run across a completely elementary argument that you somehow missed over several decades of rigorous mathematical schooling.