Fascinating ellipse trivia

Elliptic curves aren't ellipses. I get asked about that a lot.

But that doesn't mean that ellipses aren't interesting in themselves. In particular, I recently came across two interesting facts about them. The first is that an ellipse is the sum of two counter-rotating circles. The second is that if you square an ellipse that's centered at the origin you get another ellipse. I found both of those to be a bit surprising. Neither are really very hard to see.

In the first case, let's parameterize an ellipse as

x = a cos t

y = b sin t

and write

r1 = (a + b) / 2

r2 = (a b) / 2

so that

a = r1 + r2

b = r1r2

If we use complex numbers to make calculations easier, we have that

a cos t + i b sin t = (r1 + r2) cos t + i (r1r2) sin t

= r1 (cos t + i sin t) + r2 (cos t i sin t)

= r1 eit + r2 eit

which is the sum of the positively-rotating circle r1 eit and the negatively-rotating circle r2 eit. It’s also an ellipse with foci at ±√(a2 b2) = ±√(r1r2).

Now if we square that ellipse we get

(r1 eit + r2 eit)2 = r12 e2it + r22 e-2it + 2r1r2

That's another ellipse. The terms r12 e2it + r22 e-2it give us an ellipse with foci at ±2r1r2, so adding 2r1r2 just shifts the foci to 0 and 4r1r2.

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