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In the case of a arbitrary circle overlapping with an ellipse in the origin. I don't think there is a closed expression to find the intersection points. However, I was wondering, is there a (well?) known closed expression (this may be a series expansion) to describe the arc length (red) of the part of the circle overlapping with an ellipse?

Let us define the ellipse as

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

and the circle

$${(x-x_0)^2}+ {(y-y_0)^2} = r^2$$

I think the polar coordinates will be perhaps more helpful for this problem :

$$\begin{array}{lcl}x&=&r\cos(t) + x_0\\y&=&r\sin(t) + y_0\end{array}$$

enter image description here

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  • $\begingroup$ Do you know the coordinates of the centre of the circle and its radius? $\endgroup$
    – Matteo
    Aug 5, 2019 at 12:15
  • $\begingroup$ I have added the parameterizations with all parameters $\endgroup$
    – tgoossens
    Aug 5, 2019 at 12:20

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You can eliminate $y$, and get a single equation in $x$, which is is a degree-4 polynomial.
Expand the second equation, and use the first equation to eliminate the $y^2$ term. So you have $y$ equal to a quadratic in $x$. Square both sides, and eliminate $y^2$ again.

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  • $\begingroup$ Yes. But is your suggestion then that I for example use a perturbation series to describe the solutions of this polynomial? If that is what you are suggesting I can accept this answer $\endgroup$
    – tgoossens
    Aug 5, 2019 at 13:09
  • $\begingroup$ I thought one equation in one variable was simpler than two in two. I'm not sure about perturbation series, unless you are talking about Newton's method. $\endgroup$
    – Empy2
    Aug 5, 2019 at 13:24

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