Friday, April 1, 2011

Ellipse, again

Michael Starbird is a professor at UT Austin and quite well-known for mathematics teaching (here, here). I've seen (parts of) a bunch of videos he made for The Teaching Company. I like the videos that I got through pretty well, but found them a bit slow-developing. You have to love his energy and enthusiasm though.

Since I was working with ellipses the other day (here), I was reminded of a demonstration he shows in one of the videos, attributed to ____. (Unfortunately, I don't have the videos now, and I've forgotten the name. He's a French mathematician from the early 19th century. If you know, please speak up in comments).

Start with a cylinder, shown in cross section in the graphic (in black).

The two blue spheres just fit inside the cylinder, but the cylinder is tall enough that the two spheres don't touch, there is some distance between them. We take a cross section of the cylinder (shown in red), and the cross-section is level, so that the two points of the cross-section at each horizontal position are at the same height (the cross-section's extreme bottom and top are at the extreme left and right). The cross-section is an ellipse.

Alternatively, start with an ellipse whose minor axis is the same length as the radius of the cylinder and wedge it tightly inside, then add the spheres so they just touch the ellipse.

We claim that the points where the spheres touch the ellipse are the foci of the ellipse.

That's pretty remarkable. Here is the proof.

In the next figure below, consider any point on the ellipse.

On the left, we see the point together with the two foci, and a line drawn from our point to one of the foci. On the right is the cross-section again (foci not shown).

We said that this left focus is the point where the ellipse touches the lower sphere. That is, the line to the focus is a tangent to the sphere. All the tangents to the sphere from this point on the ellipse form a circle. The second tangent of interest is the perpendicular dropped down the surface of the cylinder, shown in the right panel. This line is the same length as line to the focus.

This equality holds for any point on the ellipse.

Finally, this is true for both spheres. The last figure illustrates that the sum of the perpendicular tangents for any point is a constant. Thus, the points where the spheres touch the ellipse are its foci, because the sum of the distances to any point on the ellipse, which is equal to the sum of the vertical tangents, is a constant.

Simply stunning.