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.

enter image description here

Any help in finding the parametric equations or even the methodology for finding such surfaces would be much appreciated! :)

  • $\begingroup$ As you've observed, you need a surface of constant curvature $0$ for the unrolling (or developing) into a plane to work. But there is another class of flat ruled surfaces, the so-called tangent developables. Take the set of tangent lines to a space curve. These form a ruled surface with a cuspidal edge along the original curve. This surface has none of the symmetry you're taking for granted. I'm not quite sure what's actually happening in this picture. $\endgroup$ – Ted Shifrin Aug 18 '17 at 15:51
  • $\begingroup$ See the Wikipedia article on developable surfaces: A hyperboloid is ruled but not developable. $\endgroup$ – Joseph O'Rourke Aug 19 '17 at 0:01

Shall attempt an answer as it appears to me. The "rollers" are identical shaped as 1-sheet hyperboloids, both right handed or both left handed.

The rulings that are force squeezed by heavy metal rollers placed opposite each other are asymptotic lines of a stretched helicoid surface of paper or aluminum foil, transferred from generators of hyperboloid.

A flat sheet is drawn out/rolled out to an (average) shape of a helicoid. If paper sheet is used, there would be corrugations and if thin metal foil is used plastic deformation occurs. Lines at the end of ribbon undergo a differential permanent tensile strain. The twisted helicoid that emerges is not isometric to a $K=0$ flat sheet that goes into the rollers.

Parametrization is the usual $ (x,y,z)= (u \cos v, u \sin v , c\, v) $ where $c$ relates to skew angle of the straight generators.

The above is deformation picture in the tangent plane. I suppose the surface shown inside the forming rollers is representation of corresponding normal surface, orthogonal to the deformed tangential surface.


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