More generally, a quadric surface is the locus of the points P whose coordinates satisfy the equation:
a_{200}x^2 + a_{110}xy + a_{020}y^2 + a_{011}yz + a_{002}z^2 + a_{101}xz + a_{100}x + a_{010}y + a_{001}z + a_{000} = 0
and it is straightforward to check that this condition is equivalent to
\bold{P}^TQ\bold{P} = 0, \qquad \text{where} \quad Q = \left( \begin{array}{cccc} a_{200} &\frac{1}{2}a_{110} &\frac{1}{2}a_{101} &\frac{1}{2}a_{100} \\<br />
\frac{1}{2}a_{110} &a_{020} &\frac{1}{2}a_{011} &\frac{1}{2}a_{010} \\<br />
\frac{1}{2}a_{101} &\frac{1}{2}a_{011} &a_{002} &\frac{1}{2}a_{001} \\<br />
\frac{1}{2}a_{100} &\frac{1}{2}a_{010} &\frac{1}{2}a_{001} &a_{000} \end{array} \right )
In this equation, P denotes the homogeneous coordinate vector of P.
