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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?

cosmo5
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Zhipu 'Wilson' Zhao
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  • The 3 by 3 matrix is not usually the one you consider when you have an ellipse or a hyperbola. In these cases the conic has a center and translating to that, the equation is of the form $$\begin{pmatrix}x-x_c&y-y_c\end{pmatrix} \begin{pmatrix}a_{11}&a_{12}/2\a_{12}/2&a_{22}\end{pmatrix} \begin{pmatrix}x-x_c\y-y_c\end{pmatrix}+c’=0$$ and the 2 by 2 matrix there is the one the eigenvalues for, lead to semiaxes lengths – Jan-Magnus Økland May 01 '25 at 09:15
  • Remember, we're interested in the cone section $5x^2+4xy-4y^2+7xz-4yz+24z^2=0,z=1.$ To show with an example that generally the 3 by 3 matrix gives something else, consider the cone $5x^2+4xy-4y^2+7xz-4yz+24z^2=0,$ which is what has matrix $\begin{pmatrix}-5&2&7/2\2&-4&-2\7/2&-2&24\end{pmatrix},$ transforming it, the eigenvalues of our 3 by 3 matrix give the cone $24.52438516688924x'^2-7.074937249683702y'^2-2.449447917205516z'^2=0,$ but now we've transformed away from the plane $z=1$ we're cutting with! Cutting with $z'=1$ even gives a hyperbola! – Jan-Magnus Økland May 01 '25 at 12:14

1 Answers1

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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 $z$.

Victor Liu
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  • 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 – Zhipu 'Wilson' Zhao Oct 27 '13 at 06:28