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:

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

    “Louise Bryant: I’m sorry if you don’t believe in mutual independence and free love and respect.
    Eugene O’Neill: Don’t give me a lot of parlor socialism that you learned in the village. If you were mine, I wouldn’t share you with anybody or anything. It would be just you and me. You’d be at the center of it all. You know it would feel a lot more like love than being left alone with your work.”
    —Warren Beatty (b. 1937)

    “Everything that explains the world has in fact explained a world that does not exist, a world in which men are at the center of the human enterprise and women are at the margin “helping” them. Such a world does not exist—never has.”
    —Gerda Lerner (b. 1920)