Fascinating ellipse trivia
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
r1 = (a + b) / 2
r2 = (a – b) / 2
a = r1 + r2
b = r1 – r2
If we use complex numbers to make calculations easier, we have that
a cos t + i b sin t = (r1 + r2) cos t + i (r1 – r2) sin t
= r1 (cos t + i sin t) + r2 (cos t – i sin t)
= r1 eit + r2 e–it
which is the sum of the positively-rotating circle r1 eit and the negatively-rotating circle r2 e–it. It’s also an ellipse with foci at ±√(a2 – b2) = ±√(r1r2).
Now if we square that ellipse we get
(r1 eit + r2 e–it)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.