Rotation of axes

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Rotation of Axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counter-clockwise direction with respect to the origin (0, 0) through a scalar quantity denoted by θ.

With the exception of the degenerate cases, if a general second-degree equation has a $B x y$ term, then $Ax^2\ +\ Bxy\ +\ Cy^2\ +\ Dx\ +\ Ey\ +\ F\ =\ 0$ represents one of the 4 conic sections, namely, the circle, ellipse, hyperbola, and the parabola.

1 Rotation of Loci
2 Elimination of the xy-Term by the Rotation Formula
3 Derivation of the Rotation Formula
4 Identifying Rotated Conic Sections
5 See also

[edit] Rotation of Loci

If a locus is defined on the xy-coordinate system as $\left(x,\ y\right)$ , then it is denoted as $\left(x^\prime\cos \theta\ -\ y^\prime\sin \theta,\ x^\prime\sin \theta\ +\ y^\prime\cos \theta\right)$ on the rotated x'y'-coordinate system.

[edit] Elimination of the xy-Term by the Rotation Formula

For a general, non-degenerate second-degree equation $Ax^2\ +\ Bxy\ +\ Cy^2\ +\ Dx\ +\ Ey\ +\ F\ =\ 0$ , the $B x y$ term can be removed by rotating the xy-coordinate system by an angle $θ$ , where

$\cot 2\theta\ =\ \frac{A\ -\ C}{B}$ .

[edit] Derivation of the Rotation Formula

$Ax^2\ +\ Bxy\ +\ Cy^2\ +\ Dx\ +\ Ey\ +\ F\ =\ 0,\ B \ne\ 0$ .

Now, the equation is rotated by a quantity $θ$ , hence

$A\left(x^\prime\cos \theta\ -\ y^\prime\sin \theta\right)^2\ +\ B\left(x^\prime\cos \theta\ -\ y^\prime\sin \theta\right)\left(x^\prime\sin \theta\ + y^\prime\cos \theta\right)\ +\ C\left(x^\prime\sin \theta\ +\ y^\prime\cos \theta\right)^2$

$\ +\ D\left(x^\prime\cos \theta\ -\ y^\prime\sin \theta\right)\ +\ E\left(x^\prime\sin \theta\ +\ y^\prime\cos \theta\right)\ +\ F\ =\ 0$

Expanding, the equation becomes

$A{x^\prime}^2\cos ^2\theta\ -\ 2Ax^\prime y^\prime\sin \theta\cos \theta\ +\ Ay^\prime\sin ^2\theta\ +\ B{x^\prime}^2\sin \theta\cos \theta\ +\ Bx^\prime y^\prime\cos ^2\theta$

$\ -\ Bx^\prime y^\prime\sin ^2\theta\ -\ {y^\prime}^2\cos ^2\theta\ +\ Dx^\prime\cos \theta\ -\ y^\prime\sin \theta\ +\ Ex^\prime\sin \theta\ +\ Ey^\prime\cos \theta\ +\ F\ =\ 0$

Collecting like terms,

${x^\prime}^2\left(A\cos ^2\theta\ +\ B\sin \theta\cos \theta\ +\ C\sin ^2\theta\right)\ +\ x^\prime y^\prime\left\{B\left(\cos ^2\theta\ -\ \sin ^2\theta\right)\ - 2\left(A\ -\ C\right)\left(\sin \theta\cos \theta\right)\right\}$

$\ +\ {y^\prime}^2\left(A\sin ^2\theta\ -\ B\sin \theta\cos \theta\ +\ C\cos ^2\theta\right)\ +\ x^\prime\left(D\cos \theta\ +\ E\sin \theta\right)$

$\ +\ y^\prime\left(-D\sin \theta\ +\ E\cos \theta\right)\ +\ F\ =\ 0$

In order to eliminate the x'y'-term, the coefficient of the x'y'-term must be set equal to 0.

$\begin{matrix}B\left(\cos ^2\theta\ -\ \sin ^2\theta\right)\ -\ 2\left(A\ -\ C\right)\sin \theta\cos \theta\ &=& 0 \\ \\ B\cos 2\theta\ -\ \left(A\ -\ C\right)\sin 2\theta &=& 0 \\ \\ B\cos 2\theta &=& \left(A\ -\ C\right)\sin 2\theta \\ \\ \cos 2\theta &=& \frac{\left(A\ -\ C\right)\sin 2\theta}{B} \\ \\ \cot 2\theta &=& \frac{A\ -\ C}{B} \end{matrix}$

[edit] Identifying Rotated Conic Sections

A non-degenerate conic section with the equation $Ax^2\ +\ Bxy\ +\ Cy^2\ +\ Dx\ +\ Ey\ +\ F\ =\ 0$ can be identified by evaluating the value of $B^2\ -\ 4AC$ :

$\begin{cases}\mbox{An Ellipse or a Circle},\ \mbox{if}\ B^2\ -\ 4AC\ <\ 0 \\ \mbox{A Parabola},\ \mbox{if}\ B^2\ -\ 4AC\ =\ 0 \\ \mbox{A Hyperbola},\ \mbox{if}\ B^2\ -\ 4AC\ >\ 0\end{cases}$