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If I have an ellipse centered at the origin and know the length of $a$ and $b$ and was given the length of an arc, how can I find the angle that is between the two radius from the center of the ellipse to the two end points of the arc?

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Solving this problem involves the incomplete elliptic integral of the first kind and the inverse function. The result is a rather complicated formula. Nevertheless, on a practical viewpoint, those functions are implemented in the mathematical softwares, making the numerical computation less complicated than it looks at first sight.

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