Paper model

Comment

Comment by Mårten Nettelbladt on April 23, 2012 at 11:39am

Thanks for your comments Max, this is really interesting.

Untill now I was sure my model above was actually a little double curved, I had to use a lot of force and glue to make the paper match the metal rods (you can notice some slight buckling). But when seeing your rhino model I realize you are quite right: It can be perfectly developable.
(Below are some sketches from 1999 where I try to figure this out)

Comment by Max Suica on April 23, 2012 at 3:02am

I tried the approach in rhino:

This is what the surface looks like developed onto the plane:


Now I wonder what the piece of paper in your picture looks like unrolled.

I also tried to imitate your picture using circle segments for the guide curves (blue), and also G2 continous guide curves (grey). Perhaps not surprisingly, the real paper surface looks a lot more 'natural':

  Thanks for the inspiration!

Comment by Max Suica on April 22, 2012 at 10:20pm

In this case, the two curves are equivalent to two planar smooth curves p and q on the planes z = 0 and z = 1, respectively.

I've been working the same question you pose, and after looking at your pic, I think I've come up with a sufficient condition for a smooth developable surface to exist between p and q.
 

Condition: p and q can be split into a sequence of pairs of curves (p1, q1) ..., (pn, qn) such that  each pk and qk
* have no inflection points
* turn in the same direction 
* have parallel starting and end ending tangents
* turn the same number of times

However, now I see that in your construction, these  splits happen at the points of inflection of the rail curves, but the tangents don't match. It seems like a conical section 'takes up the slack', as it were.

Awesome pic!