A surface is bounded by four lines parametrised as $(x,y,z)=$

$$ (u,0,-1), (-1<u<1); \, (0,u,1), (-1<u<1); $$

$$(\cos v, \sin v, 2 v/ \pi), (- \pi/2 < v< \pi/2); \, (-\cos v, -\sin v, 2v /\pi), (-\pi/2,< v < \pi/2); \,$$ It is required to find parametrization for constant $K$ surfaces whose

1. K= -1
2. K= 0
3. K= +1

Mean curvature H =0 for a helicoid, but is of varying K.

Untwisted surfaces catenoid, cylinder, sphere of constant $K$ are formed across two concentric circular tubes of radius 1, rotated on x-axis.They have respectively ODE as

$$ \kappa_1 + \kappa_ 2 = T $$

where T can take $ -1, 0, +1 $ values.Their twisted surface parametrization is now sought, thanks for your help.