What is an Ellipse?

There are 3 different ways to describe an ellipse:

from Wikipedia ellipse article

• The conic section created when a plane passes through a cone as shown to the right,
• The set of all points (in a plane) whose distances to two fixed points (the foci, plural of focus) add up to a constant, or
• The points (x,y) that satisfy  (x-h)²/a²+(y-k)²/b² =1.

How do we know these are the same thing?  Well, in pre-calculus courses (and sometimes intermediate algebra), there are generally proofs that the second definition leads to the third (like this one). But the connection to the conic section is always stated and never proved. For years, I've found that annoying, but hadn't looked into it until recently. I may be working with conics just a bit in my intermediate algebra course, and will be discussing them in my calc II course, so I'm brushing up this weekend.

Last year, I found a lovely proof that the first definition of the ellipse is the same as the second. It involves something called Dandelin's Spheres. It's so simple and elegant, I was surprised I'd never seen it before. Everywhere I've found it online shows the proof for ellipses, and most mention that the proofs for parabolas and hyperbolas are quite similar. I couldn't work those out on my own, though, so I went searching online again. Here's a cute rendering that does show the parabola and hyperbola, but it's still not helping me see the relationships. I'll have to sleep on this, I think.