Given the following coordinate system:
$q_1=\frac{x^2-y^2}{2}$
$q_2=xy$
$q_3=z$
What kind of surfaces are given by $q_1$ constant, $q_2$ constant, and $q_3$ constant respectively?
I am pretty sure that these surfaces are not cilindrical hyperbolics, but something weirder.
By the way, the problem is with the variables $q_1$ and $q_2$. It is clear that if $q_3$ is constant then we have a plane.
These surfaces have a name?