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:

    “Thine age asks ease, and since thy duties be
    To warm the world, that’s done in warming us.
    Shine here to us, and thou art everywhere;
    This bed thy center is, these walls, thy sphere.”
    —John Donne (1572–1631)

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

    “I don’t think America’s the center of the world anymore. I think African women will lead the way [in] ... women’s liberation ... The African woman, she’s got a country, she’s got the flag, she’s got her own army, got the navy. She doesn’t have a racism problem. She’s not afraid that if she speaks up, her man will say goodbye to her.”
    —Faith Ringgold (b. 1934)