Higher Dimensional Ellipsoids

An ellipse is a set of points in the plane which satisfy a certain equation, namely,

$\frac{(x - x_0)^2}{r_1} + \frac{(y - y_0)^2}{r_2} = k^2$

In this sense, an ellipse appears to just be a squashed circle. But there are dozens of ways to think about ellipses – as conic sections, for example – and in this post the idea I’d like to use is that of foci:

You can make an ellipse with two pins and a piece of string tied in a circle. Stick the two pins into a page and loop the string over them. Stretch the string out tight with a pen and draw all around. If you make sure to keep the string taut, you will have drawn an ellipse.

It’s a common enough geometric construction, and I’m sure many readers would know of it. I myself remember my dad showing it to me when I was a little boy. What it tells us is that an ellipse can be though of as the set of points whose total distance from two distinct points, the foci, is constant (plus the distance between those two points, but that’s always constant).

Naturally, a mathematician asks: “How can I generalise this idea?”