# Relation between ellipse general and parametric equation

I am familiar with the fact that one can relate the eigenvectors and corresponding eigenvalues of an ellipse's quadratic equation matrix, to the pose of a circle in 3-space.

When say quadratic representation, I refer to:

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

Which can be written as $[x y]^TQ[x y] = 0$

However, I have an ellipse on the following parametric representation: $$X(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi \\ Y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi$$

Now to my question. Is there any straight forward algorithm for going from parametric to quadratic representation. Or are there any papers out there, that you are familiar of, that uses the parametric representation for the pose estimation problem?

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Solved by this document from cornell: cs.cornell.edu/cv/OtherPdf/Ellipse.pdf –  Nicke Nov 7 '13 at 20:16