Is there another way to make a non developable surface developable by using plane-polygons like triangles, squares etc.?

Question

I know that a non-developable surface (like sphere, ellipsoid, paraboloid, hyperboloid,hyperbolic paraboloid helicoid, catenoid etc.) can't be flattened onto a plane without distortion (like stretching & contraction) & it has zero Gaussian curvature. Thus it is not possible to truly make a non-developable surface developable.

Therefore a non-developable surface can be approximated by a surface consisting of flat faces as plane polygons like triangles, quadrilaterals or other polygons of as minimum size as possible for best approximation such that there must be no change in the shapes & dimensions of the flat faces when transformed to the desired surface. This makes a non-developable surface as a polyhedron & a polyhedron is developable surface i.e. it can be easily flatted onto a plane without distorting any of its flat faces just by unfolding flat faces at their mating edges. Thus, a non-developable surface can be made develpable.

One of the best methods of doing so is dividing a curved surface into tiny plane-triangles called Surface Triangulation as shown in figure below

My question: Is there any other method to make a non-developable surface developable besides using net of triangles, quadrilaterals or n-polygons?

Any help will be greatly appreciated.

Answer 1

Other developable surfaces are cylinders, cones and other less known. See https://en.wikipedia.org/wiki/Developable_surface.

You can approximate your non-developable surface with patches of those.

Answer 2

In my geometry studies I have found that some shapes can be divided into hyperbolic paraboloids, which in turn are developable in a practical sense, if not in a formal sense. Consider this ruled torus produced by a trefoil knot (joined to itself with a shift of $4\pi/5$).

(3,2) torus knot with making ruled surface by self-adjoining lines

If $F(t)$ is the trefoil knot space curve, then then the torus $G(s,t)$ is defined as $G(s,t)=(1-s)F(t)+s F(t+4\pi/5)$. The domain is a rectangle with $(s,t)$ in $[0,1] \times [0,2 \pi]$. Divide the rectangle into strips like

$[0,1]\times[2\pi k/n,2\pi(k+1)/n]$ with $k$ in $0,1...n-1$.

Now each of these strips is easily approximated by a hyperbolic paraboloid containing the four corners of the strips' image. You can untwist these paraboloids by defining an axis down the middle of each and then mapping the lengths of tangent vectors in $dG$ back into the euclidean plane. The resultant figures tend to look something like this:

Hourglass shaped figure as rotation-adjusted projection of paraboloid

I cut them out of ripstop fabric and then affix them to a knot made from carbon fiber rod. The ripstop doesn't stretch, but it does twist very well back into the shape of the hyperbolic parabolid. The end model is quite authentic.