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Rotation of Conic Sections

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

If the conic section with the equation

$\displaystyle ax^2 + bxy + cy^2 + dx + ey +f = 0 $

is transformed by a rotation about the origin through the angle $ \alpha ,$ then the equation in the new coordinates $ \tilde{x}$ and $ \tilde{y}$ has the form

$\displaystyle \tilde{a}\tilde{x}^2 + \tilde{c}\tilde{y}^2 + \tilde{d}\tilde{x} + \tilde{e}\tilde{y} + \tilde{f} = 0\,, $

provided

$\displaystyle \tan 2\alpha = \frac{b}{a-c} \quad (a \neq c)$

or

$\displaystyle \alpha = \frac{\pi}{4} \quad (a = c)\ .$

The coordinate transformation which expresses the old coordinates in terms of the new ones is given by

$\displaystyle x= \tilde{x}\cos \alpha - \tilde{y}\sin \alpha \,,\quad y= \tilde{x}\sin \alpha + \tilde{y}\cos \alpha \,. $

The parameters of the new equation are

\begin{displaymath} \begin{array}{rclcrclcrcl} \tilde{a} &=& a \cos^2 \alpha +... ...ha - d \sin \alpha \,,&\ & \tilde{f} &=& f\,.\\ \end{array} \end{displaymath}

Note that in the new equation there is no mixed quadratic term, i.e. the coeffcient $ \tilde{b}$ of $ \tilde{x}\tilde{y}$ is zero. Thus this equation may be easily further transformed into a normal form of the conic section by completing squares.

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  automatically generated 3/28/2008