Matrix Representation of Conic Sections - Reduced Equation

Reduced Equation

The reduced equation of a conic section is the equation of a conic section translated and rotated so that its center lies in the center of the coordinate system and its axes are parallel to the coordinate axes. This is equivalent to saying that the coordinates are moved to satisfy these properties. See the figure.

If and are the eigenvalues of the matrix A33, the reduced equation can be written as

\lambda_1 x'^2 + \lambda_2 y'^2 + \frac{\det A_Q}{\det A_{33}} = 0

Dividing by we obtain a reduced canonical equation. For example, for an ellipse:

\frac{{x'}^2}{a^2} + \frac{{y'}^2}{b^2} = 1.

From here we get a and b.

The transformation of coordinates is given by:

T: RS(O,X,Y) \mapsto (O'=S,X',Y') \ \stackrel{\mathrm{def}}{=}\ \left\{\begin{align} \vec t &= \overrightarrow {OO'} = S\\ \alpha &= \operatorname{arccos} \frac{\vec a_1 \cdot {1 \choose 0}}{|\vec a_1|} \end{align} \right.

Read more about this topic:  Matrix Representation Of Conic Sections

Famous quotes containing the words reduced and/or equation:

    “The newspapers are the ruling power. Any other government is reduced to a few marines at Fort Independence. If a man neglects to read the Daily Times, government will go down on its knees to him, for this is the only treason these days.”
    —Henry David Thoreau (1817–1862)

    “Jail sentences have many functions, but one is surely to send a message about what our society abhors and what it values. This week, the equation was twofold: female infidelity twice as bad as male abuse, the life of a woman half as valuable as that of a man. The killing of the woman taken in adultery has a long history and survives today in many cultures. One of those is our own.”
    —Anna Quindlen (b. 1952)