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I might need some help on this problem. I searched online and didn’t find any way to solve it.

Given two points, I need to find the center of the two ellipses of given radius $rx, ry$ that intersects at those two points.

I know the two A and B point coordinates, as well as ellipses radius $rx$ and $ry$ (on $X-$axis and $Y-$ axis respectively).

Is there a way, from those $4$ variables, to calculate the center $C$ and $C'$ of each ellipse?

Here is a little diagram I made to make things clearer.

enter image description here

Nicolas Goudry
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2 Answers2

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Scale the $y$-axis until $rx=ry$.
Fit two circles.
Undo the scaling.

Empy2
  • 50,853
  • Could you please elaborate? I don’t understand how should I scale $y$-axis untir $rx = ry$.

    If I take A 50, 50; B 90, 70; $rx = 40$ and $ry = 20$, how could I apply your solution? Sorry, I’m not very good at this, so if you could try to keep it simple… Thanks

     – Nicolas Goudry Mar 02 '21 at 12:56
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If you divide all coordinates by $r_x$ and $r_y$ (respectively), the ellipses become two unit circles, by two points. The centers are on the mediatrix of the two points, at a distance from the middle given by Pythagoras (if the distance between the points is $d, \sqrt{1-\dfrac{d^2}4}$). You can compute this by vector analysis.

  • If I understand correctly, I should divide $A_x$ and $B_x$ by $r_x$, Same for $A_y$ and $B_y$ by $r_y$. If I take $A = 50; 50$, $B = 90; 70$, $r_x = 40$ and $r_y = 20$, I find $A' = 1,25; 2,5$ and $B' = 2,25; 3,5$. Sorry but I don’t understand where this is going… Could you please elaborate? – Nicolas Goudry Mar 02 '21 at 13:01