Rotation of axes

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A rotation of axes is a form of Euclidean transformation in which the entire xy-coordinate system is rotated in the counterclockwise 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 Bxy term, then Ax^2\ +\ Bxy\ +\ Cy^2\ +\ Dx\ +\ Ey\ +\ F\ =\ 0 represents one of the 3 conic sections, namely, an ellipse, hyperbola, or parabola.

Rotation of loci[edit]

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

Elimination of the xy term by the rotation formula[edit]

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

\cot 2\theta\ =\ \frac{A\ -\ C}{B} or \tan 2\theta\ =\ \frac{B}{A\ -\ C},

i.e.:

\theta\ =\ \frac{1}{2} \arctan \frac{B}{A\ -\ C}.

If A\ =\ C then \theta\ =\ \frac{\pi}{4}.

Derivation of the rotation formula[edit]

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

Now, the equation is rotated by a quantity \theta, 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\ +\ A{y^\prime}^2\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\ -\ B{y^\prime}^2\sin \theta\cos \theta\ +\ C{x^\prime}^2\sin ^2\theta\ +\ 2Cx^\prime y^\prime\sin \theta\cos \theta\ +\ C{y^\prime}^2\cos ^2\theta
\ +\ Dx^\prime\cos \theta\ -\ Dy^\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.

If A\ -\ C\neq\ 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}

If A\ -\ C\ =\ 0

\begin{matrix}B\left(\cos ^2\theta\ -\ \sin ^2\theta\right) &=& 0 \\ \\
B\cos 2\theta &=& 0 \\ \\
2\theta &=& \frac{\pi}{2} \\ \\
\theta &=& \frac{\pi}{4}\end{matrix}

Identifying rotated conic sections[edit]

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}

See also[edit]