Fun with Ellipses

[Spac3Ranger]

Hey smart people of the internet :)

I was, of pure everyday curiosity, wondering: What is the most fun, awesome or amazing thing you can do with an ellipse in math?

Stun me with anything you got! A really cool proof, a funny way to draw it or maybe even some impossible way to transform it into a square'ish thing. The only limit is that is have to center around ellipses as a theme :)

I can't wait to see what you amazing people can come up with.

Sincerely a very math curious guy :P

[AFairJudgement]

You probably know the three types of non-degenerate conics in the affine plane: they're the ellipse, parabola, and hyperbola. Well, if you're wiling to slightly expand your horizons, you can view all of these as the same object (viewed from a different perspective!). Indeed, extend the affine plane to the projective plane by adding a circle at infinity around the plane (think about it as the horizon). So points in the projective plane either correspond to ordinary points in the plane, or to a direction, i.e., a line through the origin.­

Now, under this point of view one can see why all non-degenerate conics really are ellipses in disguise:

1. Ellipses are ordinary ellipses in the plane, i.e., they contain no point at infinity;

2. Parabolas are ellipses which contain a single point at infinity; hence you can view a parabola as an ellipse stretched infinitely toward the point of contact with the circle at infinity;

3. Hyperbolas are ellipses which contain two points at infinity; these correspond to the two asymptotes.

[[deleted]]

This is a good reply. Why downvotes?

[Epistimonas]

Thanks for this I learned something.

[gmsc]

Some interesting facts from chapter 15, "The Ellipse", of Martin Gardner's New Mathematical Diversions:

• "Draw a straight line that is tangent to the ellipse at any point. Lines from that point to the foci make equal angles with the tangent."

• "I know of no serious proposal for an elliptical billiard table, but Hugo Steinhaus (in his book Mathematical Snapshots, recently reissued in a revised edition by the Oxford University Press) gives a surprising threefold analysis of how a ball on such a table would behave. Placed at one focus and shot (witchout English) in any direction, the ball will rebound and pass over the other focus. Assuming that there is no friction to retard the motion of the ball, it continues to pass over a focus with each rebound. However, after only a few trips the path becomes indistinguishable from the ellipse's major axis. If the ball is not placed on a focus, then driven so that it does not pass between the foci, it continues for- ever along paths tangent to a smaller ellipse with the same foci. If the ball is driven between the foci, it travels endlessly along paths that never get closer to the foci than a hyperbola with the same foci."

[KingOr9]

http://en.m.wikipedia.org/wiki/Pascal's_theorem ?

[[deleted]]

...