Finding ellipse-ellipse intersections in $\mathbb R^2$

David Eberly worked out the (formidable) details in these documents:

David Eberly, "The Area of Intersecting Ellipses." Geometric Tools. 2016. PDF1 download.
David Eberly, "Intersection of Ellipses." 2019. PDF2 download.

He says, "The formulas are in closed form, thus providing the exact area in terms of real-valued arithmetic." His calculations require solving quartics.

          Eberly


Algebraically - no. Your problem is reduced to computing the area of intersection of an ellipse with a disk. There can be different configurations, but one of them is when the centers coincide, and the ellipse has 4 points of intersection with the circle. Then this intersection is broken into two circular sectors and two elliptic sectors. The area of an elliptic sector is not an algebraic function (this was proved by Newton!). However there is a formula for this area in terms of transcendental functions:

https://math.stackexchange.com/questions/114371/deriving-the-area-of-a-sector-of-an-ellipse

Using this formula, you can derive a formula for your problem, but it is going to be complicated and not algebraic.

Newton's proof is so remarkable that I will reproduce it here: Let the rectilinear sides of the sector be $a$ and $b$ with common point $O$. Keep $a$ fixed and rotate $b$ about $O$. Then the area will be multi-valued function (infinitely valued!) of $b$: every time when $b$ makes a full turn, the area of the whole ellipse is added. But an algebraic function cannot be infinitely valued.

Tags:

Projective Geometry

Ag.Algebraic Geometry

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