# How can I find the equation of the shape formed by the intersection of two ellipses?

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

Manu

• 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. – Parcly Taxel Jul 2 '19 at 15:10
• You should tell us more precisely what you intend to do because there are several ways to describe this shape. – user65203 Jul 2 '19 at 15:16
• By the way, it is not enough to know the foci. – user65203 Jul 2 '19 at 15:17
• @ParclyTaxel: with the given geometry, finding the two intersection points is very simple. – user65203 Jul 2 '19 at 15:18

## 1 Answer

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

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

• Daoust : Thanks for your reply – manoos Jul 2 '19 at 15:40
• Is there any single equation with multiple parameters to describe this shape(I saw an equation of hyperellipse with 3 foci) ? – manoos Jul 3 '19 at 0:46
• @ManuFrancis: if you don't tell me the purpose, I can't answer. – user65203 Jul 3 '19 at 5:49
• 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 – manoos Jul 5 '19 at 2:23
• @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… ?) – user65203 Jul 5 '19 at 7:02