Totally quasi-umbilic timelike surfaces in ${\mathbb{R}}^{1,2}$

For a regular surface in Euclidean space ${\mathbb{R}}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2=K$; moreover, it is well-known that the only totally umbilic surfaces in ${\mathbb{R}}^3$ are planes and spheres. But for timelike surfaces in Minkowski space ${\mathbb{R}}^{1,2}$, it is possible to have $H^2=K$ at a non-umbilic point; we call such points {\em quasi-umbilic}, and we give a complete classification of totally quasi-umbilic timelike surfaces in ${\mathbb{R}}^{1,2}$.