If L is a linear transformation of (x,y,z) space, and T is a constant vector, then the transformation A(x,y,z)=L(x,y,z)+T is called an "affine transformation" of space. Similarly, if l is a linear transformation of the (u,v) plane and t is a constant vector in the plane, then a(u,v)=l(u,v)+t is an affine transformation of the plane. It is easy to see that if P(u,v) is a quadratic parametrization as above, then A(P(a(u,v))) is also a quadratic parametrization  the domain variables are transformed by a, then mapped by P into space, and then transformed by A.
The parametric maps P and Q are "equivalent" means that they are related by invertible affine transformations of domain and target, so Q(u,v)=A(P(a(u,v))). The equivalence classification of parametric maps was found by J. Peters and U. Reif in a 1997 paper. They showed that a map from R^{2} to R^{3} must be equivalent to one of fifteen types, or else its image is contained in some plane.
If the image of P is contained in a plane, then, as shown by the paper, there are fifteen types whose image is not contained in a line, five nonconstant types whose image is contained in a line, and one type where P is a constant map whose image is a point. There are also five classes of maps from R^{2} to R^{4} whose image is not contained in three dimensions, and one map R^{2} to R^{5} whose image is not contained in four dimensions, in a class by itself. So, there are a total of 1+5+15+15+5+1=42 types of maps.
For the proof, see the paper of Peters and Reif, The 42 equivalence classes of quadratic surfaces in affine nspace, which appeared in Computer Aided Geometric Design (5) 15 (1998), 459473.
The fifteen types of nonplanar maps R^{2} to R^{3} are listed in the illustrated table below, labeled by the PetersReif (PR) code.

PR Type 311.
Quadric surface: elliptic paraboloid. Implicit equation: xy^{2}z^{2}=0. Singularities: none, the image is a smooth graph. 


PR Type 312.
Quadric surface: hyperbolic paraboloid. Implicit equation: xyz=0. Singularities: none, the image is a smooth graph. 


PR Type 313.
Quadric surface: parabolic cylinder. Implicit equation: xy^{2}=0. Singularities: none, the image is a smooth graph. 


PR Type 321a.
Steiner surface: Type 5 Implicit equation: x^{2}2xz^{2}2xy2z^{2}y+z^{4}+y^{2}=0. Singularities: a double line terminating at a pinch point. The double line of the Type 5 Steiner surface that contains two pinch points and the Tshaped triple point is at infinity. 


PR Type 321b.
Quadric surface: parabolic cylinder. Implicit equation: xz^{2}=0. Singularities: a line of critical points in the domain is mapped to a fold along a parabola in the image. The parametric image does not cover the whole cylinder, only the intersection of the cylinder with the closed halfspace where y is nonnegative. Note how the parametric curves are more closely spaced on the edge. The parametric map is twotoone, except along the singular locus which is mapped to the edge. 


PR Type 321c.
Steiner surface: Type 6. Implicit equation: xy^{2}+2z^{2}yz^{4}=0. Singularities: none, the image is a smooth graph. The "cross cup" surface, with its double line at infinity. 


PR Type 322a.
Cubic Steiner surface: Type 8. Implicit equation: y^{2}xz^{2}=0. Singularities: a double line terminating at a pinch point. The Whitney Umbrella surface. 


PR Type 322b.
Cubic Steiner surface: Type 9. Implicit equation: xzz^{3}y=0. Singularities: none, the image is a smooth graph. Cayley's ruled cubic, with the double line at infinity. 


PR Type 322c.
Quadric surface: parabolic cylinder. Implicit equation: xz^{2}=0. Singularities: a line of critical points in the domain is mapped to a single point. The image is the entire cylinder, except the yaxis, one of the generating lines, where only the origin is in the image. 


PR Type 323.
Steiner surface: Type 4. Implicit equation: y^{2}+xz^{2}z^{4}=0. Singularities: a double line terminating at a pinch point. The double line of the Type 4 surface that contains the Tshaped triple point is at infinity. There is a surface in this class that has an interesting Euclidean locus property. 


PR Type 331a.
Quadric surface: cone. Implicit equation: xyz^{2}=0. Singularities: one critical point in the domain is mapped to the cone's vertex. The parametric map is otherwise twotoone: P(u,v)=P(u,v). The parametric image is half the quadric cone, including the vertex. 


PR Type 331b.
Steiner surface: Type 5. Implicit equation: 2xz^{2}+x^{2}y^{2}2x^{2}y2xz^{2}y+x^{2}+z^{4}=0. Singularities: two double lines, each terminating in a pinch point, but along one line, the selfintersection of the image is not transverse and this line also contains the pinch point of the other line  an unstable configuration. The other two pinch points of the Type 5 Steiner surface are at infinity. 


PR Type 331c.
Steiner surface: Type 1. Implicit equation: x^{2}y^{2}2x^{2}y+y^{2}2xz^{2}y8zxy+x^{2}2xz^{2}2xy+z^{4}2z^{2}y2xy^{2}=0. Singularities: three double lines meeting at a triple point, each ending in a pinch point. Steiner's Roman surface, with the other three pinch points at infinity. 


PR Type 332a.
Steiner surface: Type 6. Implicit equation: x^{3}x^{2}y^{2}+2xz^{2}yz^{4}=0. Singularities: one double line terminating at a pinch point, but the selfintersection is not transverse. The "cross cup" surface, with one pinch point at infinity. 


PR Type 332b.
Steiner surface: Type 3. Implicit equation: x^{3}x^{2}y^{2}2x^{2}+2xy^{2}+2xz^{2}y+4xz^{2}+2z^{2}yz^{4}+xy^{2}=0. Singularities: a double line terminating at a pinch point. The "cross cap" surface, with one pinch point at infinity. 