Definitions

rotation-inversion axes

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(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.

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

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{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 > 0end{cases}

See also

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