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I have 3 random points in an ellipse. Is it possible to find the radius of the ellipse?

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No. Since there are two circles containing every three points not no a line, you can take any ellipse which isn't a circle and three points on it, then there will be a circle (i. e. another ellipse) through it having different radius. –  martini May 2 '12 at 12:40
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If your ellipse is axis-aligned, you need four points to uniquely determine it; if not axis-aligned, you need five points. Your problem as it stands is underdetermined. –  J. M. May 2 '12 at 12:45
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@martini: That's not right. Three points not on a line determine a unique circle. –  TonyK May 2 '12 at 13:02
    
@TonyK Upps ... you are right of course. –  martini May 2 '12 at 13:04
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1 Answer

The equation of an ellipse whose major and minor axes are parallel to the Cartesian axes is:

$$ \left(\frac{x-x_{0}}{a}\right)^{2}+\left(\frac{y-y_{0}}{b}\right)^{2}=1 $$

As you can see, the equation contains 4 different variables. Therefore, 3 points aren't sufficient to uniquely identify the ellipse.

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