The Veronese variety is defined by the homogeneous parametric equation:
[u0: u1: u2] -> [-u0^2:u1^2:u2^2:u1*u2:u0*u2:u0*u1]
When this equation is projected into three dimensional space, it is called the Steiner surface, of which there are several varieties. I shall be focussing on the so-called Roman variety. In this case, the equation becomes:
solving for z, we get:
In order for this equation to avoid imaginary numbers and dividing by zero (in other words, to remain in the third dimension), we must have:
Similarly, we can find equivalent solutions by solving for x and y.