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Centers of Central Quadric Surfaces

1. Ellipsoids, Spheres, Cones and Hyperboloids are called Central Quadric Surfaces as they have a Geometrical Center Point across which the Surfaces are Symmetric, i.e Any Plane passing through this Point divides the Surfaces in 2 Equal Halves.
2. The Implicit Equation for any Central Quadric Surface having its Center at the Origin is given as
   
   \(Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + K=0\)   ...(1)
   
   On Translating the Center of the Quadric Surface given in equation (1) to a Point (\(x_c,y_c,z_c\)) (which is same as Translating the Equation of the Quadric Surface by (\(x_c,y_c,z_c\))), the Implicit Equation of the Quadric Surface gets updated as
   
   \(A{(x-x_c)}^2 + B{(y-y_c)}^2 + C{(z-z_c)}^2 + D(x-x_c)(y-y_c) + E(x-x_c)(z-z_c) + F(y-y_c)(z-z_c) + K = 0\)
   
   \(\Rightarrow A(x^2 + {x_c}^2 - 2x_cx) + B(y^2 + {y_c}^2 - 2y_cy) + C(z^2 + {z_c}^2 - 2z_cz) + D(xy - y_cx - x_cy + x_cy_c) + E(xz - z_cx - x_cz + x_cz_c) + F(yz - z_cy - y_cz + y_cz_c)+ K = 0\)
   
   \(\Rightarrow Ax^2 + A{x_c}^2 - 2Ax_cx + By^2 + B{y_c}^2 - 2By_cy + Cz^2 + C{z_c}^2 - 2Cz_cz + Dxy - Dy_cx - Dx_cy + Dx_cy_c + Exz - Ez_cx - Ex_cz + Ex_cz_c + Fyz - Fz_cy - Fy_cz + Fy_cz_c K = 0\)
   
   \(\Rightarrow Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + (- 2Ax_c - Dy_c - Ez_c)x + (- 2By_c - Dx_c - Fz_c)y + (- 2Cz_c - Ex_c - Fy_c)z + A{x_c}^2 + B{y_c}^2 + C{z_c}^2 + Dx_cy_c + Ex_cz_c + Fy_cz_c + K = 0\)   ...(2)
   
   \(\Rightarrow Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + K_1 = 0\)   ...(3)
   
   where
   
   \(G=- 2Ax_c - Dy_c - Ez_c\)   ...(4)
   
   \(H=- 2By_c - Dx_c - Fz_c\)   ...(5)
   
   \(I=- 2Cz_c - Ex_c - Fy_c\)   ...(6)
   
   \(K_1=A{x_c}^2 + B{y_c}^2 + C{z_c}^2 + Dx_cy_c + Ex_cz_c + Fy_cz_c + K\)   ...(7)
   
   The equations (2) and (3) above give the Equation of the Central Quadric Surface whose Center is Translated by an Offset \((x_c,y_c,z_c)\) from the Origin.
   
   Now, the equations (4) , (5) and (6) above can be written in form of a Matrix Equation as follows
   
   \(\begin{bmatrix}-2A & -D & -E\\-D & -2B & - F\\-E & -F & - 2C\end{bmatrix}\begin{bmatrix}x_c\\y_c\\z_c\end{bmatrix}=\begin{bmatrix}G\\H\\I\end{bmatrix}\)   ...(8)
   
   The Coordinates of Center of the Quadric Surface \((x_c,y_c,z_c)\) can be determined by solving the System of Linear Equations given by the Matrix Equation (8).