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}/a^{2}+(y-k)^{2}/b^{2}=1.

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.

I think there is a fourth way. What about the locus of points whose distance from a fixed point is proportional to the distance to a line, where the ratio is less than 1?

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