When a sheet is drawn taut from a cylindrical roll of paper, a flat sheet traveling in a straight line results. When a sheet is drawn from a cone, the paper unfurls out as a flat disk. So, how about unfolding out a sheet from a hyperboloid (that is, a hyperboloid of one sheet, not to confuse by what I mean by "sheet")? To be more precise, what are the parametric equations that define such a surface? In all cases, the shapes from which the sheets are rolled can freely rotate about their axis of revolution.
Since a hyperboloid is a ruled surface, it is possible to draw a sheet from it in the first place without distortion of the sheet, i.e. isometry. One thing I notice is that a hyperboloid has non-zero Gaussian curvature, and so the sheet cannot be unfolded into a plane. So perhaps what it means to unfold out a sheet tautly isn't well defined for non-zero Gaussian curvature scenarios.
This image I found suggests the answer is indeed a helicoid. However it doesn't give away much in the way of parametric equations, or whether this holds for the general case.
Any help in finding the parametric equations or even the methodology for finding such surfaces would be much appreciated! :)