Question

Suppose you have a 3 dimensional object, represented as a 3d mesh in some common file format. How would you devise an algorithm to decompose the mesh into one or more 2d ‘nets’ – that is, a 2-dimensional representation that can be cut out and folded to create the original 3d object.

Amongst other things, the algorithm would need to account for:

• Multiple possible decompositions for any given object
• Handling fitting a mesh into fixed size canvases (sheets of paper).
• Recognizing when two panels in the net would overlap (and are thus invalid).
• Breaking a mesh up into multiple nets if they can’t be represented as a single one, due to overlap or page size constraints.
• Generating tabs in the appropriate places, for attaching adjacent faces.

The obvious degenerate case is simply to create one net per face, with tabs on half the edges. This isn’t ideal, obviously: The ideal case is a single continuous net. The reality for complex shapes is likely to be somewhere in the middle.

I realize that finding the optimal net (fewest nets / least pages) is probably computationally expensive, but a good heuristic for finding ‘good enough’ nets would suffice.

Answer 1

When I read your question, the words “automatic papercraft algorithm” came to me. So I googled it and found Papercraft Models using Generalized Cylinders (pdf) by Massarwi et al.

We propose a new method for producing
unfolded papercraft patterns of
rounded toy animal figures from
triangulated meshes by means of
strip-based approximation. Although in
principle a triangulated model can be
unfolded simply by retaining as much
as possible of its connectivity while
checking for intersecting triangles in
the unfolded plane, creating a pattern
with tens of thousands of triangles is
unrealistic. Our approach is to
approximate the mesh model by a set of
continuous triangle strips with no
internal vertices. Initially, we
subdivide our mesh into parts
corresponding to the features of the
model. We segment each part into zonal
regions, grouping triangles which are
similar topological distances from the
part boundary. We generate triangle
strips by simplifying the mesh while
retaining the borders of the zonal
regions and additional cut-lines. The
pattern is then created simply by
unfolding the set of strips. The
distinguishing feature of our method
is that we approximate a mesh model by
a set of continuous strips, not by
other ruled surfaces such as parts of
cones or cylinders. Thus, the
approximated unfolded pattern can be
generated using only mesh operations
and a simple unfolding algorithm.
Furthermore, a set of strips can be
crafted just by bending the paper
(without breaking edges) and can
represent smooth features of the
original mesh models.

There is also an earlier related paper called Paper craft models from meshes (9MB pdf) by Shatz et al.

This paper introduces an algorithm for
segmenting a mesh into developable
approximations. The algorithm can be
used in various applications in CAD
and computer graphics. This paper
focuses on paper crafting and
demonstrates that the algorithm
generates approximations that are
developable, easy to cut, and can be
glued together. It is also shown that
the error between the given model and
the paper model is small.

Source: http://www.ee.technion.ac.il/~ayellet/images/sel-papers-pic-5.jpg