In Homogeneous coordinates (x_1, x_2, x_3), the equation of a circle C is a(x_1^2 + x_2^2) + 2f x_2 x_3 + 2g x_1 x_3 + c x_3^2 = 0. The discriminant of this circle is defined as Δ = left bracketing bar a | 0 | g 0 | a | f g | f | c right bracketing bar = a(a c - f^2 - g^2), and the quadratic form q(C) = a c - f^2 - g^2 is the basic invariant.

conic section discriminant