I was wondering if anyone knows of any research where an ellipse is used as an irrational number generator.

$\pi$ is equal to the ratio of a circle's circumference to its diameter so it would make sense that you could generate other irrational numbers such that they're greater than $1$ but less than $\pi$ such as $e$ or $\phi$ by taking the circumference of an ellipse and dividing it by its major radius multiplied by $2$.

I'm sure there's been work on this before but can anyone point me in the right direction?

Would it also be possible to generate all irrational numbers greater than $\pi$ by dividing an ellipses perimeter by its minor radius multiplied by $2$?

  • $\begingroup$ The sort of equations you get is $ \pi E(s, \epsilon) = e ... \pi $ where E is an elliptic integral , $\epsilon $ is its eccentricity and $s$ is arc length. $\endgroup$ – Narasimham Dec 18 '16 at 2:10
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    $\begingroup$ Any number between $2$ and $\pi$, whether rational or irrational, can be achieved as the circumference of an ellipse with major axis $1$ and suitably chosen minor axis. However, finding the minor axis for a desired circumference, or vice versa, is nontrivial: en.wikipedia.org/wiki/Ellipse#Circumference $\endgroup$ – Rahul Dec 18 '16 at 2:18
  • $\begingroup$ Perhaps you should explain what you mean by "generate" here. Are you looking for a 'formula' for a given irrational number in terms of the circumference and the major axis? $\endgroup$ – Winther Dec 18 '16 at 2:36
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    $\begingroup$ Well then it's answered by Rahul's comment above. We have $\frac{C}{2a} = 2\int_0^{\pi/2}\sqrt{1-\epsilon^2\sin^2\theta}{\rm d}\theta$ and this integral is bounded below by $2$ and above by $\pi$. By continuity any number inbetween can be attained. You can understand this geometrically: if the ellipse is maximally streched it's a line ($C=4a$). If it's minimally streached it's a circle ($C = 2\pi a$). $\endgroup$ – Winther Dec 18 '16 at 2:44
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    $\begingroup$ I think the interesting thing about $\pi$ in a circle is that from a rational radius you can get an irrational circumference. As Rahul wrote and Winther confirmed, you can get any number in that range as such a ratio, but to achieve that the ratio of the major and minor axis, or the eccentricity or whatever shape parameter you choose would be irrational as well for most irrational numbers. In my opinion this makes the whole approach pretty uninteresting, as turrning one irrational ratio into another one isn't that special. $\endgroup$ – MvG Dec 18 '16 at 12:00

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