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

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

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

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

The Dini surface does not meet requirement of a helix border for case 1. Neither Mean curvature H =0 helicoid of varying K satisfies this case 1.
 
Untwisted constant $ H$  CMC surfaces catenoid, cylinder, sphere of soap films form across two concentric circular tube edges of radius 1 rotated on x-axis.They have respectively their ODE connecting principal curvatures as:
  
$$ \kappa_1  +  \kappa_ 2 = T,$$

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

EDIT 1:

As far as spanning between two helical lines is concerned, I have chosen circular toroidal upper,lower and flat points in the WolframAlpha link in place of Elliptic Integrals/functions as an approximation to the needed situation, K > 0, K = 0, K <0 regions of the torus can be seen.

EDIT2

A helical cuspidal boundary defined by David Brander as a pseudospherical front must have a spherical front' or gorge/groove etc. imho,that at  is how one views positive and negative constant  curvature surfaces..David Hilbert proved a cuspidal edge boundary and the conoidal point vertex for $K\equiv _1$, and for spherical case $K\equiv =+1$ someone else  also proved  the same...Am I right? 

EDIT3:

Asked in SE Math for a perhaps easier formulation where the limit / boundary of quadrangle patch is removed defining  an infinitely long curved strip:

[Infinite Warped Strip][1]

[ Const K surfaces between helices][2]


  [1]: http://math.stackexchange.com/questions/1829592/helocoidal-surface-of-gauss-curvature-1
  [2]: https://www.wolframalpha.com/input/?i=ParametricPlot3D%5B%7B(Cos%20%5Bt%20%5D%2B5)%20Cos%20%5Bv%5D,(%20Cos%5B%20t%20%5D%20%2B5%20)%20Sin%20%5Bv%5D,%20Sin%20%5Bt%20%5D%2B.34%20v%20%7D,%7B%20t,0,2%20Pi%7D,%7Bv,0,%205Pi%7D%5D