A quadric surface is defined by 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, which can be expressed as \bold{P}^TQ\bold{P} = 0, where Q is a symmetric matrix constructed from the coefficients. The subscripts in the coefficients indicate the number of factors of x, y, and z that multiply the respective coefficient, such as a_{011} representing the coefficient of x^0y^1z^1. Understanding this notation is crucial for interpreting quadric surfaces in computer vision.
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More generally, a quadric surface is the locus of the points P whose coordinates satisfy the equation:
[tex]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[/tex]
and it is straightforward to check that this condition is equivalent to
[tex]\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 )[/tex]
In this equation, P denotes the homogeneous coordinate vector of P.