Folding Paper

by Chris Wilkins

Mar 4, 2010

I'm bending a quadrilateral piece of paper along one diagonal. I've modeled a rigid version, but I want to make a surface that most closely approximates what a piece of paper would be shaped like, using the same 4 edges, but without creasing the diagonal fold. When folding it by hand, it just appears as a strip of paper that is twisting.

So I'm looking for advice on how to make the paper surface. (holding area of paper the same)

See images and definition file below, if you have any ideas. Thanks.

Here are videos of the piece that is made up of these surfaces:



  • up

    Tomohiro Tachi

    You might be interested in the results of our paper "(Non)existence of Pleated Folds" presented at JCCGG 2009, which proves non-existence of the non-triangulated hypar model (and the existence of triangulated models).

    One of the theorems we have proved is that a piece of developable surface bounded by straight crease lines is polygonal (piecewise linear) in a half-folded state.
    A smooth surface between ABCD must always have negative Gaussian curvature like Srf4Pt.
    4

    Mar 4, 2010

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      Mårten Nettelbladt

      I would make a large paper model and study it very closely.
      Do the sides of the paper remain straight or do they curve?
      Can you find any straight "lines" in the the curved paper shape?

      It is quite possible to connect those 4 points (A,B,C,D) with a developable surface if you don't have to follow the straight edge lines (AB,BC,CD,DA).

      Can you post a photo of a (fairly large) paper model that is showing the desired "twist"?
      It just needs to be a sheet of paper that you hold in your hand.
      1

      Mar 5, 2010

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        Mårten Nettelbladt

        Hi Chris, I tried to replicate your model.

        (1) First test is with loose edges.

        (2) The edges start to curve quite a lot to deal with the twist:
        (Cone sections are seen, with centre points in A, B and C)


        (3)


        (4)Then I made the edges rigid (like in your model):


        (5) The paper deals with this in two ways, one is crumpling:


        (6) The other way is curved edges:


        (7) The curved edges allow for a developable shape that consits of two cone sections (joined by the green line, which is also the location of your extra fold.) Centre points are A and C.


        I guess crumpling is tricky to do in Grasshopper (perhaps something for Kangaroo?),
        so I would use the two cone sections to create an accurate model in GH.

        Or, actually, stick with the extra fold! (A folded surface is also developable, and you've already made lots of other folds).

        Good luck!
        1

        Mar 5, 2010