Matrix Representation of Conic Sections - Axes

Axes

The major and minor axes are two lines determined by the center of the conic as a point and eigenvectors of the associated matrix as vectors of direction.

a_{1,2} \ \stackrel{\mathrm{def}}{=}\ \left\{\begin{matrix} S(x_0,y_0) &\qquad \mbox{(center of the conic)}\\ \vec u(u_x,u_y) &\qquad \mbox{(eigenvector of }A_{33}) \end{matrix} \right.

So we can write a canonical equation:

a_{1,2} \ \stackrel{\mathrm{def}}{=}\ \frac{x-x_0}{u_x} = \frac{y-y_0}{u_y}

Because a 2x2 matrix has 2 eigenvectors, we obtain 2 axes.

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