For continuous shearlet systems the parabolic scaling matrices $$A_a := \begin{pmatrix} a & 0 \\ 0 & \sqrt{a}\end{pmatrix}, \quad a > 0$$ are very important. Why are they called parabolic?
I thought of making a quadratic form from the matrix, i.e. $Q(x,y) := (x,y) \cdot A \cdot (x,y)^T$. This gives $$ Q(x,y) = ax^2 + \sqrt{a}x y, $$ which has discriminant $b^2 - 4ac = - 4 a^{\frac{3}{2}} < 0$, so we should be calling it hyperbolic, right?