Rapid prototyping with paper and a printer

[Kragen Sitaker]

(A quick first draft from some time ago.  Needs references.)

I saw a web page for a piece of software called "Pepakura", which
takes 3-D models as input and prints out a colored cut-out pattern on
paper, which you can then cut out and assemble to get an approximation
of the original 3-D model.  There are uncolored tabs on the cutout
that merely serve to hold together seams.  It seems to be relatively
limited, in that the models are polyhedral and don't have any kind of
continuous curvature, just edges and vertices.  I think it's possible
to do better.

I have also seen stuffed-animal models of monsters from
first-person-shooter video games, reportedly made from those 3-D
models themselves, turned into cloth patterns, cut out, sewed, and
stuffed.

Certainly you can make cones; indeed, just bringing two intersecting
straight edges of a piece of paper together makes a cone, whatever the
shape of the paper.  There's even more freedom than that, though.

Suppose you have a paper disc and you cut a pie wedge out of it, then
bring the edges together.  Now you have a cone --- it has (infinite)
positive curvature at its tip, and zero curvature everywhere else.

Suppose, instead that your pie wedge edges are curved like a pair of
parentheses which have been tilted so their tops touch, so the edges
are nearly parallel at the edge of the pie, but not very sharply
curved in the middle.  Now when you bring these edges together, the
cone is more acutely pointed: it has negative curvature everywhere but
the point.  Note also that this is difficult to do and puts stress on
the paper elsewhere, since the paper doesn't naturally have negative
curvature anywhere except at the closed wound.

And if you curve the edges of the pie wedge inward instead --- so that
the end of the wedge is very sharply pointed, but the edges curve
outward to follow the remaining crust of the pie as they approach it
--- your cone has positive curvature, looking more like a hemisphere.

All of these shapes are somewhat floppy, because you have only
determined the curvature of this small part of the surface; this
leaves the shape of the overall surface underdetermined.  When your
surface forms a closed curve, its shape is no longer underdetermined,
and it becomes rigid.  This happens gradually as it approaches being a
closed surface.

In all of these deformed cone shapes, the paper resists quite a bit.
Paper is a somewhat different medium from cloth in this way.  Cloth is
mostly happy to assume negative or positive curvature by stretching
along the diagonals of its weave, up to some fairly large limits, but
paper is not so compliant.  I think a good paper-modeling program will
need to understand this.

I'm curious what the limits, or benefits, of the everyday laser or
inkjet printer are for making 3-D shapes.  It *would* be nice to have
a texture on your object to start with.

Perhaps you could print a pattern on paper, waxed the paper, cut out
the pattern, closed the seams (which can be done easily with a little
heat when the paper is sufficiently waxed, and then undone and redone
if a bad job was done), filled the resulting shape with cement mixed
with sand, let the cement set, and cut off the waxed paper.  Clearly
you can do most of this without the printer if you can do it at all
--- what value does the printer add?  Perhaps it could give you a
better idea of the eventual shape of the thing and let you make things
that fit together.  Perhaps also, software can do a better job of
minimizing the amount of cutting you have to do than you can.

Perhaps you could use various kinds of sprayable or paintable
coatings: paints, latex, concrete (not as a first layer, of course).

There's another possible way to use a laser printer for rapid
prototyping: print slices of the desired shape on some normal
laser-printable, use some technique (photoresist etching, or the
stone-tablet etching technique from the previous post, if it works
well enough) to transform the shapes into solid slices, then bolt them
together through built-in bolt holes.