 A conicoid is said to be symmetric with the point P if the conicoid is symmetric to the origin when the equation of the conicoid is transformed with origin as the center.
 The origin is the center of the conicoid if ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has u = v = w = 0.
 A conicois S given by the equation F(x, y, z) = ax^{2} + by^{2} + cz^{2} + 2fyz + 2gzx + 2gxy + 2ux + 2vy + 2wz + d = 0 has a point P(x_{0}, y_{0}, z_{0}) as the center iff
ax_{0} + hy_{0} + gz_{0} + u = 0 hx_{0} + by_{0} + fz_{0} + v = 0 gx_{0} + fy_{0} + cz_{0} + w = 0
 A conicoid has a central conicoid if it has a unique center, it is non central conicoid if it does not has a center or has more than one center.
Standard forms of central conicoids
Type  Standard form 
cone  ax^{2} + by^{2} + cz^{2} = 0 
imaginary ellipsoid  x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1 
ellipsoid  x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1 
hyperboloid of one sheet  x^{2}/a^{2} + y^{2}/b^{2}  z^{2}/c^{2} = 1 x^{2}/a^{2}  y^{2}/b^{2} + z^{2}/c^{2} = 1 x^{2}/a^{2} + y^{2}/b^{2} + z^{2}/c^{2} = 1 
hyperboloid of two sheets  x^{2}/a^{2}  y^{2}/b^{2}  z^{2}/c^{2} = 1 x^{2}/a^{2}  y^{2}/b^{2} + z^{2}/c^{2} = 1 x^{2}/a^{2} + y^{2}/b^{2}  z^{2}/c^{2} = 1 
 The condition for a line to be a tangent to the central conicoid ax^{2} + by^{2} + cz^{2} = 1 at (x_{0}, y_{0}, z_{0}) is ax_{0}α + by_{0}β + cz_{0}γ = 0, where α, β and γ are the direction ratios of the line.
 The equation of the tangent plane to a central conicoid ax^{2} + by^{2} + cz^{2} = 1 at a point (x_{0}, y_{0}, z_{0}) is axx_{0} + byy_{0} + czz_{0} = 1.
 The condition that the plane ux + vy + wz = p is a tangent to the central conicoid ax^{2} + by^{2} + cz^{2} = 1 is u^{2}/a + v^{2}/b + w^{2}/z = p^{2}
 A planar section of a central conicoid is a conic section.
