There are three questions here. We imagine a flexible membrane that is scrolled out so as to straighten it.
1) How can we find a surface isometrically mapped from a surface of constant negative Gauss curvature $K=-1$ (hyperbolic ring type contained between two cuspidal equators) so that its central minimal girth circumference of the circle maps to a straight line?
Fig 1 $ ( z= cos x cosh y ) $ is a deformed surface intuitively expected to be similar to desired deformed surface as elongated sinusoidal arcs parallel to spine appear. But the spine itself is curved, and not straight, that cannot be of correct form.
2) How can we find a surface isometrically mapped from a surface of constant negative $K= -1 $ contained between two cuspidal equators so that the central minimal girth circumference of the circle maps to an ellipse of given eccentricity? It is diametrically squeezed in the central waist region. This I feel is quite possible but again do not know how to formulate leading to its parametrization. In Fig 2 such deformed surface (of a narrow waist Vaseline bottle) with different central major and minor axes is pictured to get an idea of the surface imagined.
3) Next, how do we find mapping of a surface of constant positive K (of revolution) so that the central minimal circumference of the circle maps to a straight line? Intuitively I feel that this is impossible (physically seen, reduced cuspidal lengths tear up if forced that way) but unable to prove it to be so.
Hope it is sufficiently clear, I shall explain further on receipt of your comments. I tried to do geodesic mapping (as geodesics are invariant in isometric mappings) but taking components in a yet to be deformed system was not a successful exercise. Thanks in advance for all help.