Quadric Surface Rotation

1. Quadric Surface Rotation refers to Changing the Orientation of a Quadric Surface Object. This is done by rotating the General Quadratic Equation in 3 Variables representing the Quadric Surface.
2. The General Quadratic Equation in 3 Variables representing a Quadric Surface is given as follows
   
   \(Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz +K = 0\)   ...(1)
   
   Rotation of any given Quadric Surface represented by equation(1) above is given in terms of its Euler / Tait-Bryan Rotation Angles which are Set of Elementary Rotations around \(X\), \(Y\) and \(Z\) Axis.
   
   On rotating the equation (1) Counter Clockwise with respect to \(Z\) Axis by an Angle \(\theta\) as per the Rule of Rotation of Equations, the updated equation is given as
   
   \(A{(x\cos\theta + y \sin \theta)}^2 + B{(y \cos\theta - x \sin\theta)}^2 + Cz^2 + D(x\cos\theta + y \sin\theta)(y \cos\theta - x \sin\theta) \\ + E(x\cos\theta + y \sin\theta)z + F(y \cos\theta - x \sin\theta)z + G(x\cos\theta + y \sin\theta) + H(y \cos\theta - x \sin\theta) + Iz + K = 0\)
   
   \(\Rightarrow A(x^2 {\cos}^2\theta + y^2 {\sin}^2\theta + 2xy \sin\theta \cos\theta) + B(y^2 {\cos}^2\theta + x^2 {\sin}^2\theta - 2xy \sin\theta \cos\theta) + Cz^2 + D(xy {\cos}^2\theta - xy {\sin}^2\theta - x^2 \sin\theta \cos\theta + y^2 \sin\theta \cos\theta) \\ + E \cos\theta xz + E \sin\theta yz + F \cos\theta yz - F \sin\theta xz + G \cos\theta x + G \sin\theta y + H \cos\theta y - H \sin\theta x + Iz + K = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta - D \sin\theta \cos\theta + B {\sin}^2\theta) x^2 + (A{\sin}^2\theta + D \sin\theta \cos\theta + B {\cos}^2\theta) y^2 + Cz^2 + (2A \sin\theta \cos\theta + D{\cos}^2\theta - D{\sin}^2\theta - 2B \sin\theta \cos\theta) xy \\ + (E \cos\theta - F \sin\theta) xz + (F \cos\theta + E \sin\theta) yz + (G \cos\theta - H \sin\theta) x + (H \cos\theta + G \sin\theta) y + Iz + K = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta - D \sin\theta \cos\theta + B {\sin}^2\theta) x^2 + (A{\sin}^2\theta + D \sin\theta \cos\theta + B {\cos}^2\theta) y^2 + Cz^2 + ((A-B) \sin 2\theta + D \cos 2\theta) xy \\ + (E \cos\theta - F \sin\theta) xz + (F \cos\theta + E \sin\theta) yz + (G \cos\theta - H \sin\theta) x + (H \cos\theta + G \sin\theta) y + Iz + K = 0\)   ...(2)
   
   \(\Rightarrow A_1x^2 + B_1y^2 + Cz^2 + D_1xy + E_1xz + F_1yz + G_1x + H_1y + Iz + K \)   ...(3)
   
   where
   
   \(A_1=A{\cos}^2\theta - D \sin\theta \cos\theta + B {\sin}^2\theta\)
   
   \(B_1=A{\sin}^2\theta + D \sin\theta \cos\theta + B {\cos}^2\theta\)
   
   \(D_1=(A-B) \sin 2\theta + D \cos 2\theta\)
   
   \(E_1=E \cos\theta - F \sin\theta\)
   
   \(F_1=F \cos\theta + E \sin\theta\)
   
   \(G_1=G \cos\theta - H \sin\theta\)
   
   \(H_1=H \cos\theta + G \sin\theta\)
   
   The equations (2) and (3) above give the Equation of the Quadric Surface Rotated Counter Clockwise with respect to \(Z\) Axis by an Angle \(\theta\).
   
   Please note the Rotating a Quadric Surface with respect to \(Z\) Axis Does Not Change the Value of its Constant of the Equation or Values of its Co-efficients \(z^2\) or \(z\) . However it Changes the Values of its Quadratic Co-efficients (\(x^2, y^2 , xy, xz\) and \(yz\)) and its Linear Co-efficients (\(x\) and \(y\)).
   
   
   On rotating the equation (1) Counter Clockwise with respect to \(X\) Axis by an Angle \(\theta\) as per the Rule of Rotation of Equations, the updated equation is given as
   
   \(Ax^2 + B{(y \cos\theta + z \sin\theta)}^2 + C{(z \cos\theta - y \sin\theta)}^2 + D(y \cos\theta + z \sin\theta)x \\ + E(z\cos\theta - y \sin\theta)x + F(y \cos\theta + z \sin\theta)(z \cos\theta - y \sin\theta) + Gx + H(y \cos\theta + z \sin\theta) + I(z \cos\theta - y \sin\theta) + K = 0\)
   
   \(\Rightarrow Ax^2 + B(y^2 {\cos}^2\theta + z^2 {\sin}^2\theta + 2yz \sin\theta \cos\theta) + C(z^2 {\cos}^2\theta + y^2 {\sin}^2\theta - 2yz \sin\theta \cos\theta) + D \cos\theta xy + D \sin\theta xz \\ + E \cos\theta xz - E \sin\theta xy + F(yz {\cos}^2\theta - yz {\sin}^2\theta - y^2 \sin\theta \cos\theta + z^2 \sin\theta \cos\theta) + Gx + H \cos\theta y + H \sin\theta z + I \cos\theta z - I \sin\theta y + K = 0\)
   
   \(\Rightarrow Ax^2 + (B{\cos}^2\theta - F \sin\theta \cos\theta + C {\sin}^2\theta) y^2 + (B{\sin}^2\theta + F \sin\theta \cos\theta + C {\cos}^2\theta) z^2 + (D \cos\theta - E \sin\theta) xy \\ + (E \cos\theta + D \sin\theta) xz + (2B \sin\theta \cos\theta + F{\cos}^2\theta - F{\sin}^2\theta - 2C \sin\theta \cos\theta) yz + Gx + (H \cos\theta - I \sin\theta) y + (I \cos\theta + H \sin\theta) z + K = 0\)
   
   \(\Rightarrow Ax^2 + (B{\cos}^2\theta - F \sin\theta \cos\theta + C {\sin}^2\theta) y^2 + (B{\sin}^2\theta + F \sin\theta \cos\theta + C {\cos}^2\theta) z^2 + (D \cos\theta - E \sin\theta) xy \\ + (E \cos\theta + D \sin\theta) xz + ((B-C) \sin 2\theta + F \cos 2\theta) yz + Gx + (H \cos\theta - I \sin\theta) y + (I \cos\theta + H \sin\theta) z + K = 0\)   ...(4)
   
   \(\Rightarrow Ax^2 + B_1y^2 + C_1z^2 + D_1xy + E_1xz + F_1yz + Gx + H_1y + I_1z + K \)   ...(5)
   
   where
   
   \(B_1=B{\cos}^2\theta - F \sin\theta \cos\theta + C {\sin}^2\theta\)
   
   \(C_1=B{\sin}^2\theta + F \sin\theta \cos\theta + C {\cos}^2\theta\)
   
   \(D_1=D \cos\theta - E \sin\theta\)
   
   \(E_1=E \cos\theta + D \sin\theta\)
   
   \(F_1=(B-C) \sin 2\theta + F \cos 2\theta\)
   
   \(H_1=H \cos\theta - I \sin\theta\)
   
   \(I_1=I \cos\theta + H \sin\theta\)
   
   The equations (4) and (5) above give the Equation of the Quadric Surface Rotated Counter Clockwise with respect to \(X\) Axis by an Angle \(\theta\).
   
   Please note the Rotating a Quadric Surface with respect to \(X\) Axis Does Not Change the Value of its Constant of the Equation or Values of its Co-efficients \(x^2\) or \(x\) . However it Changes the Values of its Quadratic Co-efficients (\(y^2, z^2 , xy, xz\) and \(yz\)) and its Linear Co-efficients (\(y\) and \(z\)).
   
   
   On rotating the equation (1) Counter Clockwise with respect to \(Y\) Axis by an Angle \(\theta\) as per the Rule of Rotation of Equations, the updated equation is given as
   
   \(A{(x\cos\theta - z \sin \theta)}^2 + By^2 + C{(z \cos\theta + x \sin\theta)}^2 + D (x\cos\theta - z \sin\theta)y\\ + E(x\cos\theta - z \sin\theta)(z \cos\theta + x \sin\theta) + F(z \cos\theta + x \sin\theta)y + G(x\cos\theta - z \sin\theta) + Hy + I(z \cos\theta + x \sin\theta) + K = 0\)
   
   \(\Rightarrow A(x^2 {\cos}^2\theta + z^2 {\sin}^2\theta - 2xz \sin\theta \cos\theta) + By^2 + C(z^2 {\cos}^2\theta + x^2 {\sin}^2\theta + 2xz \sin\theta \cos\theta) + D \cos\theta xy - D \sin\theta yz \\ + E(xz {\cos}^2\theta - xz {\sin}^2\theta + x^2 \sin\theta \cos\theta - z^2 \sin\theta \cos\theta) + F \cos\theta yz + F \sin\theta xy + G \cos\theta x - G \sin\theta z + Hy + I \cos\theta z + I \sin\theta x + K = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta + E \sin\theta \cos\theta + C {\sin}^2\theta) x^2 + By^2 + (A{\sin}^2\theta - E \sin\theta \cos\theta + C {\cos}^2\theta) z^2 + (D \cos\theta + F \sin\theta) xy \\ + (2C \sin\theta \cos\theta + E{\cos}^2\theta - E{\sin}^2\theta - 2A \sin\theta \cos\theta) xz + (F \cos\theta - D \sin\theta) yz + (G \cos\theta + I \sin\theta) x + Hy + (I \cos\theta - G \sin\theta) z + K = 0\)
   
   \(\Rightarrow (A{\cos}^2\theta + E \sin\theta \cos\theta + C {\sin}^2\theta) x^2 + By^2 + (A{\sin}^2\theta - E \sin\theta \cos\theta + C {\cos}^2\theta) z^2 + (D \cos\theta + F \sin\theta) xy \\ + ((C-A) \sin 2\theta + E \cos 2\theta) xz + (F \cos\theta - D \sin\theta) xz + (G \cos\theta + I \sin\theta) x + Hy + (I \cos\theta - G \sin\theta) z + K = 0\)   ...(6)
   
   \(\Rightarrow A_1x^2 + By^2 + C_1z^2 + D_1xy + E_1xz + F_1yz + G_1x + Hy + I_1z + K \)   ...(7)
   
   where
   
   \(A_1=A{\cos}^2\theta + E \sin\theta \cos\theta + C {\sin}^2\theta\)
   
   \(C_1=A{\sin}^2\theta - E \sin\theta \cos\theta + C {\cos}^2\theta\)
   
   \(D_1=D \cos\theta + F \sin\theta\)
   
   \(E_1=(C-A) \sin 2\theta + E \cos 2\theta\)
   
   \(F_1=F \cos\theta - D \sin\theta\)
   
   \(G_1=G \cos\theta + I \sin\theta\)
   
   \(I_1=I \cos\theta - G \sin\theta\)
   
   The equations (6) and (7) above give the Equation of the Quadric Surface Rotated Counter Clockwise with respect to \(Y\) Axis by an Angle \(\theta\).
   
   Please note the Rotating a Quadric Surface with respect to \(Y\) Axis Does Not Change the Value of its Constant of the Equation or Values of its Co-efficients \(y^2\) or \(y\) . However it Changes the Values of its Quadratic Co-efficients (\(x^2, z^2 , xy, xz\) and \(yz\)) and its Linear Co-efficients (\(x\) and \(z\)).
   
   
   Also note that Clockwise Rotation of the Quadric Surface can be done by replacing \(\theta\) with \(-\theta\) in all the equations above.