Conic Section Rotation

1. Conic Section Rotation refers to Changing the Orientation of a Conic Section Object. This is done by rotating the General Quadratic Equation in 2 Variables representing the Conic Section.
2. The General Quadratic Equation in 2 Variables representing a Conic Section is given as follows
   
   \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)   ...(1)
   
   On rotating the above equation (1) Counter Clockwise by an Angle \(\theta\) as per the Rule of Rotation of Equations, the updated equation is given as
   
   \(A{(x\cos\theta + y \sin \theta)}^2 + B(x\cos\theta + y \sin\theta)(y \cos\theta - x \sin\theta) + C{(y \cos\theta - x \sin\theta)}^2 + D(x\cos\theta + y \sin\theta) + E(y \cos\theta - x \sin\theta) + F = 0\)
   
   \(\Rightarrow A(x^2 {\cos}^2\theta + y^2 {\sin}^2\theta + 2xy \sin\theta \cos\theta) + B(xy {\cos}^2\theta - xy {\sin}^2\theta - x^2 \sin\theta \cos\theta + y^2 \sin\theta \cos\theta) + C(y^2 {\cos}^2\theta + x^2 {\sin}^2\theta - 2xy \sin\theta \cos\theta) \\ + D \cos\theta x + D \sin\theta y + E \cos\theta y - E \sin\theta x + F = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta - B \sin\theta \cos\theta + C {\sin}^2\theta) x^2 + (2A \sin\theta \cos\theta + B{\cos}^2\theta - B{\sin}^2\theta - 2C \sin\theta \cos\theta) xy + (A{\sin}^2\theta + B \sin\theta \cos\theta + C {\cos}^2\theta) y^2 \\ + (D \cos\theta - E \sin\theta) x + (E \cos\theta + D \sin\theta) y + F = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta - B \sin\theta \cos\theta + C {\sin}^2\theta) x^2 + ((A-C) \sin 2\theta + B \cos 2\theta) xy + (A{\sin}^2\theta + B \sin\theta \cos\theta + C {\cos}^2\theta) y^2 + (D \cos\theta - E \sin\theta) x + (E \cos\theta + D \sin\theta) y + F = 0\)   ...(2)
   
   \(\Rightarrow A_1x^2 + B_1xy + C_1y^2 + D_1x + E_1y + F \)   ...(3)
   
   where
   
   \(A_1=A{\cos}^2\theta - B \sin\theta \cos\theta + C {\sin}^2\theta\)
   
   \(B_1=(A-C) \sin 2\theta + B \cos 2\theta\)
   
   \(C_1=A{\sin}^2\theta + B \sin\theta \cos\theta + C {\cos}^2\theta\)
   
   \(D_1=D \cos\theta - E \sin\theta\)
   
   \(E_1=E \cos\theta + D \sin\theta\)
   
   The equations (2) and (3) above give the Equation of the Conic Section Rotated Counter Clockwise by an Angle \(\theta\).
   
   Please note the Rotating a Conic Section Does Not Change the Value of its Constant of the Equation. However it Changes the Values of its Quadratic Co-efficients (\(x^2, xy\) and \(y^2\)) and its Linear Co-efficients (\(x\) and \(y\)).
   
   Also note that Clockwise Rotation of the Conic Section can be done by replacing \(\theta\) with \(-\theta\) in all the equations above.