182 research outputs found
Unfolding Orthogonal Terrains
It is shown that every orthogonal terrain, i.e., an orthogonal (right-angled)
polyhedron based on a rectangle that meets every vertical line in a segment,
has a grid unfolding: its surface may be unfolded to a single non-overlapping
piece by cutting along grid edges defined by coordinate planes through every
vertex.Comment: 7 pages, 7 figures, 5 references. First revision adds Figure 7, and
improves Figure 6. Second revision further improves Figure 7, and adds one
clarifying sentence. Third corrects label in Figure 7. Fourth revision
corrects a sentence in the conclusion about the class of shapes now known to
be grid-unfoldabl
Continuous Blooming of Convex Polyhedra
We construct the first two continuous bloomings of all convex polyhedra.
First, the source unfolding can be continuously bloomed. Second, any unfolding
of a convex polyhedron can be refined (further cut, by a linear number of cuts)
to have a continuous blooming.Comment: 13 pages, 6 figure
An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures
We study algorithmic aspects of bending wires and sheet metal into a
specified structure. Problems of this type are closely related to the question
of deciding whether a simple non-self-intersecting wire structure (a
carpenter's ruler) can be straightened, a problem that was open for several
years and has only recently been solved in the affirmative.
If we impose some of the constraints that are imposed by the manufacturing
process, we obtain quite different results. In particular, we study the variant
of the carpenter's ruler problem in which there is a restriction that only one
joint can be modified at a time. For a linkage that does not self-intersect or
self-touch, the recent results of Connelly et al. and Streinu imply that it can
always be straightened, modifying one joint at a time. However, we show that
for a linkage with even a single vertex degeneracy, it becomes NP-hard to
decide if it can be straightened while altering only one joint at a time. If we
add the restriction that each joint can be altered at most once, we show that
the problem is NP-complete even without vertex degeneracies.
In the special case, arising in wire forming manufacturing, that each joint
can be altered at most once, and must be done sequentially from one or both
ends of the linkage, we give an efficient algorithm to determine if a linkage
can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry -
Theory and Application
Star Unfolding Convex Polyhedra via Quasigeodesic Loops
We extend the notion of star unfolding to be based on a quasigeodesic loop Q
rather than on a point. This gives a new general method to unfold the surface
of any convex polyhedron P to a simple (non-overlapping), planar polygon: cut
along one shortest path from each vertex of P to Q, and cut all but one segment
of Q.Comment: 10 pages, 7 figures. v2 improves the description of cut locus, and
adds references. v3 improves two figures and their captions. New version v4
offers a completely different proof of non-overlap in the quasigeodesic loop
case, and contains several other substantive improvements. This version is 23
pages long, with 15 figure
Boxelization: folding 3D objects into boxes
We present a method for transforming a 3D object into a cube or a box using a continuous folding sequence. Our method produces a single, connected object that can be physically fabricated and folded from one shape to the other. We segment the object into voxels and search for a voxel-tree that can fold from the input shape to the target shape. This involves three major steps: finding a good voxelization, finding the tree structure that can form the input and target shapes' configurations, and finding a non-intersecting folding sequence. We demonstrate our results on several input 3D objects and also physically fabricate some using a 3D printer
Any Regular Polyhedron Can Transform to Another by O(1) Refoldings
We show that several classes of polyhedra are joined by a sequence of O(1)
refolding steps, where each refolding step unfolds the current polyhedron
(allowing cuts anywhere on the surface and allowing overlap) and folds that
unfolding into exactly the next polyhedron; in other words, a polyhedron is
refoldable into another polyhedron if they share a common unfolding.
Specifically, assuming equal surface area, we prove that (1) any two
tetramonohedra are refoldable to each other, (2) any doubly covered triangle is
refoldable to a tetramonohedron, (3) any (augmented) regular prismatoid and
doubly covered regular polygon is refoldable to a tetramonohedron, (4) any
tetrahedron has a 3-step refolding sequence to a tetramonohedron, and (5) the
regular dodecahedron has a 4-step refolding sequence to a tetramonohedron. In
particular, we obtain at most 6-step refolding sequence between any pair of
Platonic solids, applying (5) for the dodecahedron and (1) and/or (2) for all
other Platonic solids. As far as the authors know, this is the first result
about common unfolding involving the regular dodecahedron
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