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I recently understood the meaning of focus points and how to abuse them for generating some fun games like Loop. (Reading pool backwards)

In Loop one uses a pool table in the shape of an ellipse and puts the hole in of its focus points. Thus one can always hole a ball by making it go through its other focus point.

I believe in 2-dimensional there are just ellipse to have this property, also that an object can't have more than 2 focus points. However I cannot prove it.

Therefore I would like to read up on that topic. I am particularly interested, which kind of properties an n-dimensional shape, table or whatever has to fulfill for it to have 2,3 or more focus points.

As always thanks in advance for any constructive comment or answer.

Imago
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I believe in $2$ dimensions there are just ellipses that have this property, also that an object can't have more than $2$ focus points.

You believe wrongly ! You can have as many foci as you want, even in $2$D, and the shape thus obtained is called an n-ellipse or a “multifocal ellipse”. My advice to you is to do what I did: Go to the Desmos page, draw two points at $(\pm1,0)$, and a third one of variable coordinates which is mobile $($you will have to use sliders$)$. Then type in the equation of the $3$-ellipse, and modify the position of the last point to inspect its behavior.

Lucian
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  • Thank you! This is indeed very interesting. I didn't expect the "math of focus points" to be so overseeable. Well, now I have something read and maybe I can come up with a nice problem. Thanks again. – Imago Aug 22 '15 at 17:22