Matrix Representation of Conic Sections - Center

Center

In the center of the conic, the gradient of the quadratic form Q vanishes, so: \nabla Q = = .

We can calculate the center by taking the first two rows of the associated matrix, multiplying each by (x, y, 1)T, setting both inner products equal to 0, and solving the system.

S \ \stackrel{\mathrm{def}}{=}\ \left\{ \begin{matrix} a_{11} + a_{12}x + a_{13}y & = & 0 \\ a_{21} + a_{22}x + a_{23}y & = & 0 \end{matrix} \right. \ \stackrel{\mathrm{def}}{=}\ \left\{\begin{matrix} D/2 + Ax + (B/2)y & = & 0 \\ E/2 + (B/2)x + Cy & = & 0 \end{matrix} \right.

This becomes

\begin{pmatrix} x_c \\ y_c \end{pmatrix} = \begin{pmatrix} A & B/2 \\ B/2 & C \end{pmatrix}^{-1} \begin{pmatrix} -D/2 \\ -E/2 \end{pmatrix} = \begin{pmatrix} (BE-2CD)/(4AC-B^2) \\ (DB-2AE)/(4AC-B^2) \end{pmatrix}

Note that in the case of a parabola, defined by (4AC-B2) = 0, there is no center since the above denominators become zero.

Read more about this topic:  Matrix Representation Of Conic Sections

Famous quotes containing the word center:

    “Children can’t be a center of life and a reason for being. They can be a thousand things that are delightful, interesting, satisfying, but they can’t be a wellspring to live from. Or they shouldn’t be.”
    —Doris Lessing (b. 1919)

    “Every beloved object is the center point of a paradise.”
    —Novalis [Friedrich Von Hardenberg] (1772–1801)

    “This is a strange little complacent country, in many ways a U.S.A. in miniature but of course nearer the center of disturbance!”
    —Eleanor Roosevelt (1884–1962)