I am trying to find the equation of shape formed by the intersection of two ellipses. I have foci of each ellipse. The pictorial representation of the problem is shown in the figure below.

Clicke to see figure: Ellipse intersection

How can I generate the equation of shape shown pink colour?

I don't have that much base in mathematics, so if my question has any problem, please provide me feedback.

Thanks in advance


  • $\begingroup$ You will have to intersect the two ellipses. This is hardly a trivial matter, but a search on MSE should bring up a good number of methods. $\endgroup$ – Parcly Taxel Jul 2 '19 at 15:10
  • $\begingroup$ You should tell us more precisely what you intend to do because there are several ways to describe this shape. $\endgroup$ – user65203 Jul 2 '19 at 15:16
  • $\begingroup$ By the way, it is not enough to know the foci. $\endgroup$ – user65203 Jul 2 '19 at 15:17
  • $\begingroup$ @ParclyTaxel: with the given geometry, finding the two intersection points is very simple. $\endgroup$ – user65203 Jul 2 '19 at 15:18

The pink area can be described by a system of inequations, for instance

$$\begin{cases} \dfrac{(x-x_1)^2}{a_1^2}+\dfrac{y^2}{b_1^2}\le 1,\\ \dfrac{(x-x_2)^2}{a_2^2}+\dfrac{y^2}{b_2^2}\le 1. \end{cases}$$

  • $\begingroup$ Daoust : Thanks for your reply $\endgroup$ – manoos Jul 2 '19 at 15:40
  • $\begingroup$ Is there any single equation with multiple parameters to describe this shape(I saw an equation of hyperellipse with 3 foci) ? $\endgroup$ – manoos Jul 3 '19 at 0:46
  • $\begingroup$ @ManuFrancis: if you don't tell me the purpose, I can't answer. $\endgroup$ – user65203 Jul 3 '19 at 5:49
  • $\begingroup$ I need to find the projection of a cylinderical hole in 2D plane by using a point source illumination at different positions. The projection of hole will be an intersection of two ellipses. This is my purpose $\endgroup$ – manoos Jul 5 '19 at 2:23
  • $\begingroup$ @ManuFrancis: I see. But what kind of representation do you need ? What kind of processing follows ? (Computing the area, filling in a raster image, flattening as a polygon… ?) $\endgroup$ – user65203 Jul 5 '19 at 7:02

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