Real space quadrics and μ-bases

In Euclidean 3-space, a quadric surface is the zero set of a quadratic equation in three variables. Its projective closure can be given as the closure of the image of a rational parametrization P:R2→R4 where P maps the parameters (s,t)∈R2(s,t)∈R2 to the tuple (a,b,c,d)∈R4(a,b,c,d)∈R4 and a, b, c, d are linearly independent quadratic polynomials, with gcd(a, b, c, d) = 1. This paper provides an algorithm to classify the type of quadric surface, and identify the normal forms solely based on the parametrization of the quadric surface.