What is the maximum number of non-overlapping circles that can be placed inside an ellipse?

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Consider a standard ellipse of equation: $$\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$$

and let the circle be given a radius $r$. Is there some algorithm useful to calculate it?

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It isn't known for an ellipse, because in the special case where $a=b,$ it is still unknown what optimal circle packings there are for circles . . .

See https://en.wikipedia.org/wiki/Circle_packing_in_a_circle