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When encountering the general equation of a conic section $$a_{11}x^2 + a_{12}xy + a_{22}y^2 + b_1x + b_2y + c = 0 $$

I can write it in matrix form as a quadratic form of the vector $(x,y,1)^T$. But what then? What should be done to reach the form of the standard equation of a conic section, i.e. of an ellipse/hyperbola/parabola with center translated and conic rotated?

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You need to compute the eigenvalue decomposition of the quadratic form, which diagonalizes the matrix. So suppose you have $$ [x,y,1] A [x,y,1]^T = 0 $$ Since $A$ can always be made symmetric, then it should have an eigenvalue decomposition $A=Q\Lambda Q^T$ where $Q$ is orthogonal and $\Lambda$ is diagonal. Then, if you use the rotated coordinates $z=Q[x,y,1]^T$, you get $$ z^T \Lambda z = 0$$ which you can multiply out to get the standard form in terms of the components of $z$.

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  • $\begingroup$ How should i deal with the third co-ordinate? The matrix A has 3 eigenvectors and which 2 should i choose? what should i do if in the rotated coordinate the 3rd coordinate is not 1? thx for answering $\endgroup$ – Zhipu 'Wilson' Zhao Oct 27 '13 at 6:28

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