Rotation of axes

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 $Bxy$ term, then $Ax^2\; +\; Bxy\; +\; Cy^2\; +\; Dx\; +\; Ey\; +\; F\; =\; 0$ represents one of the 3 conic sections, namely, the ellipse, hyperbola, and the parabola.

Rotation of loci

If a locus is defined on the xy-coordinate system as $left(x,\; yright)$, then it is denoted as $left(xcos\; theta\; +\; ysin\; theta,\; -xsin\; theta\; +\; ycos\; thetaright)$ on the rotated x'y'-coordinate system. Likewise, if a locus is defined on the x'y'-coordinate system as $left(x^prime\; ,\; y^primeright)$, then it is denoted as $left(x^primecos\; theta\; -\; y^primesin\; theta,\; x^primesin\; theta\; +\; y^primecos\; thetaright)$ on the "un-rotated" xy-coordinate system.

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 $Bxy$ term can be removed by rotating the xy-coordinate system by an angle $theta$, where

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

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 $theta$, hence

$Aleft(x^primecos\; theta\; -\; y^primesin\; thetaright)^2\; +\; Bleft(x^primecos\; theta\; -\; y^primesin\; thetaright)left(x^primesin\; theta\; +\; y^primecos\; thetaright)\; +\; Cleft(x^primesin\; theta\; +\; y^primecos\; thetaright)^2$

$+\; Dleft(x^primecos\; theta\; -\; y^primesin\; thetaright)\; +\; Eleft(x^primesin\; theta\; +\; y^primecos\; thetaright)\; +\; F\; =\; 0$

Expanding, the equation becomes

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

$-\; Bx^prime\; y^primesin\; ^2theta\; -\; B\{y^prime\}^2cos\; ^2theta\; +\; C\{x^prime\}^2sin\; ^2theta\; +\; 2Cx^prime\; y^primesin\; thetacos\; theta\; +\; C\{y^prime\}^2cos\; ^2theta$

$+\; Dx^primecos\; theta\; -\; Dy^primesin\; theta\; +\; Ex^primesin\; theta\; +\; Ey^primecos\; theta\; +\; F\; =\; 0$

Collecting like terms,

$\{x^prime\}^2left(Acos\; ^2theta\; +\; Bsin\; thetacos\; theta\; +\; Csin\; ^2thetaright)\; +\; x^prime\; y^primeleft\{Bleft(cos\; ^2theta\; -\; sin\; ^2thetaright)\; -\; 2left(A\; -\; Cright)left(sin\; thetacos\; thetaright)right\}$

$+\; \{y^prime\}^2left(Asin\; ^2theta\; -\; Bsin\; thetacos\; theta\; +\; Ccos\; ^2thetaright)\; +\; x^primeleft(Dcos\; theta\; +\; Esin\; thetaright)$

$+\; y^primeleft(-Dsin\; theta\; +\; Ecos\; thetaright)\; +\; F\; =\; 0$

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

Identifying rotated conic sections according to A. Lenard

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 $B^2\; -\; 4AC$:

mbox{A parabola}, mbox{if} B^2 - 4AC = 0 mbox{A hyperbola}, mbox{if} B^2 - 4AC > 0end{cases}

See also

• Rotation