Wykan wrote:
> 
> You know those drinking straws with the little "twisty" bit near the top?
> You can pop it down and compact it, or stretch it out straight, or bend it,
> etc. You know what I mean, yes?
> 
> Well, what would you call that shape? I mean, the shape of each of those
> little segments. It seems like it's just two flat-topped cones, yes?
> 
> But suppose (since it is true) that I want to make an object with a
> "twisty" section like that (not a straw, but that's the best analogy I
> could think of). How would I describe the way each segment changes its
> shape when bent? The "cone" is no longer even, so to speak; the angle of
> one rim is different than the other (more acute on the "inside" of the
> bend).

One way to think about a pleated material like this is as a stack of
concentric rings, of alternate sizes (small, large, small, large,
etc.).  Each ring would then be connected by a flexible material - in
the case of a plastic straw, the rings are not entirely rigid, as the
material is not that flexible.  If we then say that the small rings are
the same radius as the unpleated sections of the straw, the larger rings
would be expected to change size somewhat as the pleated section is
stretched, ie. the more stretch, the closer the larger rings get to the
size of the smaller rings.

This effect can certainly be simulated quite easily using cones; the
situation gets a little more complicated when the straw is bent into an
arc.  In this case, we could imagine a connector joining each ring at
some point (say, the centre).  Applying a little rotation to each
connector (but keeping the connector length the same) then gives an arc
with the distance between rings in the inner curve being shorter than
the centre connector, and the distance between rings in the outer curve
being longer (while the overall length of the pleated segment remains
the same).

As far as modelling this - you could perhaps use the same cones used in
the straight section, but either rotate or shear them, and then clip
them at the correct angle (so the ends are no longer perpendicular).  In
both cases, one of the rings will no longer be circular, due to
distortion (rotation would give elliptical larger rings, while shearing
would give elliptical smaller rings).  If you wanted to keep both rings
circular, you could connect the two rotated ring outlines using
calculated smooth_triangles or bicubic_patches.

Obviously, some reasonable amount of maths will be required,
particularly if you want to create animations.  I actually have used a
similar method to the above in my Object Bender include file, and if you
like I could look at extending a little on the maths used in that file
to create something that does what you describe.

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