Finding the points of a line with a known direction and distance joining 2 ellipses

Question

I have 2 ellipses, say $e_1$ and $e_2$.

I want to draw a line $l$ connecting $e_1$ and $e_2$ in a known direction $(u,v)$, with a known distance $d$.

Is there a way to solve for the points of intersection of $l$ on both $e_1$ and $e_2$? I am implementing this in software and the only way I have thought of is testing every point on $e_1$, but was thinking if there is an analytical way.

Answer 1

We can use Lagrange's multipliers with 2 constraints( the two equations for the ellipses) and the minimized function.

Answer 2

Because you know the direction $(u,v)$ of your connecting line, you can imagine a collection of parallel lines intersecting $e_1$ and $e_2$, all in direction $(u,v)$:

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It is relatively easy to compute the intersection of a line and an ellipse. E.g., here is one explication; here is another. If the intersection points are $p_1$ and $p_2$ for a line $L$, you want to solve $||p_1 - p_2|| = d$ to find the line $L$ whose intersection points are $d$ apart.