How to find center of ellipse from two points (these are just points on the ellipse, not related to foci), and two radii ($r_x$ and $r_y$, from standard definition of the ellipse $\frac{x^2}{r_x^2} + \frac{y^2}{r_y^2}=1$) of that ellipse? (there will be two centers, I assume)
Calculating center of the ellipse
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geometry
conic-sections
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0Edited the question. – 2010-11-21
1 Answers
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You seem to be asking about an ellipse whose axes are aligned with the $x$ and $y$ directions. Note that an ellipse which is rotated by an arbitrary angle is still an ellipse. But you cannot uniquely determine such an ellipse from the information given.
If two points $(x_1, y_1)$ and $(x_2, y_2)$ pass through an ellipse centred at $(x_0, y_0)$ with semi-axes along $x$ and $y$ of length $r_x$ and $r_y$ respectively, then the points $(x_1/r_x, y_1/r_y)$ and $(x_2/r_x, y_2/r_y)$ pass through a circle of unit radius with centre $(x_0/r_x, y_0/r_y)$. So if you can find the centre of this circle, you can find the centre of the corresponding ellipse.