Quadric Surface Translation

1. Quadric Surface Translation refers to Changing the Position of a Quadric Surface Object. This is done by translating 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)
   
   On translating the above equation (1) by an offset \((x_t,y_t,z_t)\) as per the Rule of Translation of Equations, the updated equation is given as
   
   \(A{(x-x_t)}^2 + B{(y-y_t)}^2 + C{(z-z_t)}^2 + D(x-x_t)(y-y_t) + E(x-x_t)(z-z_t) + F(y-y_t)(z-z_t) + G(x-x_t) + H(y-y_t) + I(z-z_t) + K = 0\)
   
   \(\Rightarrow A(x^2 + {x_t}^2 - 2x_tx) + B(y^2 + {y_t}^2 - 2y_ty) + C(z^2 + {z_t}^2 - 2z_tz) + D(xy - y_tx - x_ty + x_ty_t) + E(xz - z_tx - x_tz + x_tz_t) + F(yz - z_ty - y_tz + y_tz_t) + Gx - Gx_t + Hy - Hy_t + Iz - Iz_t + K = 0\)
   
   \(\Rightarrow Ax^2 + A{x_t}^2 - 2Ax_tx + By^2 + B{y_t}^2 - 2By_ty + Cz^2 + C{z_t}^2 - 2Cz_tz + Dxy - Dy_tx - Dx_ty + Dx_ty_t + Exz - Ez_tx - Ex_tz + Ex_tz_t + Fyz - Fz_ty - Fy_tz + Fy_tz_t + Gx - Gx_t + Hy - Hy_t + Iz - Iz_t + K = 0\)
   
   \(\Rightarrow Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + (G - 2Ax_t - Dy_t - Ez_t)x + (H - 2By_t - Dx_t - Fz_t)y + (I - 2Cz_t - Ex_t - Fy_t)z + A{x_t}^2 + B{y_t}^2 + C{z_t}^2 + Dx_ty_t + Ex_tz_t + Fy_tz_t - Gx_t - Hy_t - Iz_t + K = 0\)   ...(2)
   
   \(\Rightarrow Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + G_1x + H_1y + I_1z + K_1 = 0\)   ...(3)
   
   where
   
   \(G_1=G - 2Ax_t - Dy_t - Ez_t\)
   
   \(H_1=H - 2By_t - Dx_t - Fz_t\)
   
   \(I_1=I - 2Cz_t - Ex_t - Fy_t\)
   
   \(K_1=A{x_t}^2 + B{y_t}^2 + C{z_t}^2 + Dx_ty_t + Ex_tz_t + Fy_tz_t - Gx_t - Hy_t - Iz_t + K\)
   
   The equations (2) and (3) above give the Equation of the Quadric Surface Translated by an Offset \((x_t,y_t,z_t)\).
   
   Please note the Translating a Quadric Surface Does Not Change the Values of its Quadratic Co-efficients (\(x^2, y^2, z^2, xy, xz\) and \(yz\)). However it Changes the Values of its its Linear Co-efficients (\(x, y\) and \(z\)) and the Constant of the Equation.