Rotation of axes

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 = 0represents 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.

egin{matrix}Bleft(cos ^2 heta - sin ^2 heta ight) - 2left(A - C ight)sin hetacos heta &=& 0 \ \Bcos 2 heta - left(A - C ight)sin 2 heta &=& 0 \ \Bcos 2 heta &=& left(A - C ight)sin 2 heta \ \cos 2 heta &=& frac{left(A - C ight)sin 2 heta}{B} \ \cot 2 heta &=& frac{A - C}{B} end{matrix}

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:

:::egin{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}

ee also

*Rotation


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