105 research outputs found
Locked and Unlocked Polygonal Chains in 3D
In this paper, we study movements of simple polygonal chains in 3D. We say
that an open, simple polygonal chain can be straightened if it can be
continuously reconfigured to a straight sequence of segments in such a manner
that both the length of each link and the simplicity of the chain are
maintained throughout the movement. The analogous concept for closed chains is
convexification: reconfiguration to a planar convex polygon. Chains that cannot
be straightened or convexified are called locked. While there are open chains
in 3D that are locked, we show that if an open chain has a simple orthogonal
projection onto some plane, it can be straightened. For closed chains, we show
that there are unknotted but locked closed chains, and we provide an algorithm
for convexifying a planar simple polygon in 3D with a polynomial number of
moves.Comment: To appear in Proc. 10th ACM-SIAM Sympos. Discrete Algorithms, Jan.
199
Locked and Unlocked Polygonal Chains in 3D
In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D with a polynomial number of moves
Computational Geometry Column 39
The resolution of a decades-old open problem is described: polygonal chains
cannot lock in the plane.Comment: 4 pages, 2 figures. To appear in SIGACT News and in Int. J. Comp.
Geom. App
Locked and Unlocked Polygonal Chains in Three Dimensions
This paper studies movements of polygonal chains in three dimensions whose links are not allowed to cross or change length. Our main result is an algorithmic proof that any simple closed chain that initially takes the form of a planar polygon can be made convex in three dimensions. Other results include an algorithm for straightening open chains having a simple orthogonal projection onto some plane, and an algorithm for making convex any open chain initially configured on the surface of a polytope. All our algorithms require only O (n) basic moves.
Polygonal Chains Cannot Lock in 4D
We prove that, in all dimensions d>=4, every simple open polygonal chain and
every tree may be straightened, and every simple closed polygonal chain may be
convexified. These reconfigurations can be achieved by algorithms that use
polynomial time in the number of vertices, and result in a polynomial number of
``moves.'' These results contrast to those known for d=2, where trees can
``lock,'' and for d=3, where open and closed chains can lock.Comment: Major revision of the Aug. 1999 version, including: Proof extended to
show trees cannot lock in 4D; new example of the implementation straightening
a chain of n=100 vertices; improved time complexity for chain to O(n^2);
fixed several minor technical errors. (Thanks to three referees.) 29 pages;
15 figures. v3: Reference update
On Reconfiguring Tree Linkages: Trees can Lock
It has recently been shown that any simple (i.e. nonintersecting) polygonal
chain in the plane can be reconfigured to lie on a straight line, and any
simple polygon can be reconfigured to be convex. This result cannot be extended
to tree linkages: we show that there are trees with two simple configurations
that are not connected by a motion that preserves simplicity throughout the
motion. Indeed, we prove that an -link tree can have
equivalence classes of configurations.Comment: 16 pages, 6 figures Introduction reworked and references added, as
the main open problem was recently close
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
A 2-chain can interlock with a k-chain
One of the open problems posed in [3] is: what is the minimal number k such
that an open, flexible k-chain can interlock with a flexible 2-chain? In this
paper, we establish the assumption behind this problem, that there is indeed
some k that achieves interlocking. We prove that a flexible 2-chain can
interlock with a flexible, open 16-chain.Comment: 10 pages, 6 figure
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