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)

    “My center is giving way, my right is in retreat; situation excellent. I shall attack.”
    —Ferdinand Foch (1851–1929)

    “I am the center of the world, but the control panel seems to be somewhere else.”
    —Mason Cooley (b. 1927)