= Projective conic
{title2=$\{[x:y:z]:q(x,y,z)=0\}$}
A <projective conic> is the zero locus in a <projective plane> of a nonzero homogeneous degree-two <polynomial>. The equation is considered up to multiplication by a nonzero scalar. Over the <complex numbers>, a <symmetric matrix> for the <quadratic form> has rank three, two, or one: these give a nonsingular conic, two distinct <projective lines>, or one line counted twice. Diagonalization of a <quadratic form> and rescaling its nonzero diagonal entries prove this classification. The smooth model $XZ-Y^2=0$ is parametrized by the degree-two <Veronese map> $[s:t]\mapsto[s^2:st:t^2]$.
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