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 $left(x,\; y\; ight)$, then it is denoted as $left(xcos\; heta\; +\; ysin\; heta,\; -xsin\; heta\; +\; ycos\; heta\; ight)$ on the rotated x'y'-coordinate system.Likewise, if a locus is defined on the x'y'-coordinate system as $left(x^prime\; ,\; y^prime\; ight)$, then it is denoted as $left(x^primecos\; heta\; -\; y^primesin\; heta,\; x^primesin\; heta\; +\; y^primecos\; heta\; ight)$ 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 $heta$, where

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

Derivation of the rotation formula

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

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

$Aleft(x^primecos\; heta\; -\; y^primesin\; heta\; ight)^2\; +\; Bleft(x^primecos\; heta\; -\; y^primesin\; heta\; ight)left(x^primesin\; heta\; +\; y^primecos\; heta\; ight)\; +\; Cleft(x^primesin\; heta\; +\; y^primecos\; heta\; ight)^2$::$+\; Dleft(x^primecos\; heta\; -\; y^primesin\; heta\; ight)\; +\; Eleft(x^primesin\; heta\; +\; y^primecos\; heta\; ight)\; +\; F\; =\; 0$

Expanding, the equation becomes

$A\{x^prime\}^2cos\; ^2\; heta\; -\; 2Ax^prime\; y^primesin\; hetacos\; heta\; +\; Ay^primesin\; ^2\; heta\; +\; B\{x^prime\}^2sin\; hetacos\; heta\; +\; Bx^prime\; y^primecos\; ^2\; heta$::$-\; Bx^prime\; y^primesin\; ^2\; heta\; -\; B\{y^prime\}^2cos\; ^2\; heta\; +\; C\{x^prime\}^2sin\; ^2\; heta\; +\; 2Cx^prime\; y^primesin\; hetacos\; heta\; +\; C\{y^prime\}^2cos\; ^2\; heta$:::$+\; Dx^primecos\; heta\; -\; Dy^primesin\; heta\; +\; Ex^primesin\; heta\; +\; Ey^primecos\; heta\; +\; F\; =\; 0$

Collecting like terms,

$\{x^prime\}^2left(Acos\; ^2\; heta\; +\; Bsin\; hetacos\; heta\; +\; Csin\; ^2\; heta\; ight)\; +\; x^prime\; y^primeleft\{Bleft(cos\; ^2\; heta\; -\; sin\; ^2\; heta\; ight)\; -\; 2left(A\; -\; C\; ight)left(sin\; hetacos\; heta\; ight)\; ight\}$::$+\; \{y^prime\}^2left(Asin\; ^2\; heta\; -\; Bsin\; hetacos\; heta\; +\; Ccos\; ^2\; heta\; ight)\; +\; x^primeleft(Dcos\; heta\; +\; Esin\; heta\; ight)$:::$+\; y^primeleft(-Dsin\; heta\; +\; Ecos\; heta\; ight)\; +\; F\; =\; 0$

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 dubious

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

:::

ee also

*Rotation