Definitions

# Rotation of axes

[ak-seez]
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 $left\left(x, yright\right)$, then it is denoted as $left\left(xcos theta + ysin theta, -xsin theta + ycos thetaright\right)$ on the rotated x'y'-coordinate system. Likewise, if a locus is defined on the x'y'-coordinate system as $left\left(x^prime , y^primeright\right)$, then it is denoted as $left\left(x^primecos theta - y^primesin theta, x^primesin theta + y^primecos thetaright\right)$ 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

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

## 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

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

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

Expanding, the equation becomes

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

$- Bx^prime y^primesin ^2theta - B\left\{y^prime\right\}^2cos ^2theta + C\left\{x^prime\right\}^2sin ^2theta + 2Cx^prime y^primesin thetacos theta + C\left\{y^prime\right\}^2cos ^2theta$
$+ Dx^primecos theta - Dy^primesin theta + Ex^primesin theta + Ey^primecos theta + F = 0$

Collecting like terms,

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

$+ \left\{y^prime\right\}^2left\left(Asin ^2theta - Bsin thetacos theta + Ccos ^2thetaright\right) + x^primeleft\left(Dcos theta + Esin thetaright\right)$
$+ y^primeleft\left(-Dsin theta + Ecos thetaright\right) + F = 0$

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

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

## 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 $B^2 - 4AC$:

$begin\left\{cases\right\}mbox\left\{An ellipse or a circle\right\}, mbox\left\{if\right\} B^2 - 4AC < 0$
mbox{A parabola}, mbox{if} B^2 - 4AC = 0 mbox{A hyperbola}, mbox{if} B^2 - 4AC > 0end{cases}