2,023 research outputs found
Locked and Unlocked Chains of Planar Shapes
We extend linkage unfolding results from the well-studied case of polygonal
linkages to the more general case of linkages of polygons. More precisely, we
consider chains of nonoverlapping rigid planar shapes (Jordan regions) that are
hinged together sequentially at rotatable joints. Our goal is to characterize
the families of planar shapes that admit locked chains, where some
configurations cannot be reached by continuous reconfiguration without
self-intersection, and which families of planar shapes guarantee universal
foldability, where every chain is guaranteed to have a connected configuration
space. Previously, only obtuse triangles were known to admit locked shapes, and
only line segments were known to guarantee universal foldability. We show that
a surprisingly general family of planar shapes, called slender adornments,
guarantees universal foldability: roughly, the distance from each edge along
the path along the boundary of the slender adornment to each hinge should be
monotone. In contrast, we show that isosceles triangles with any desired apex
angle less than 90 degrees admit locked chains, which is precisely the
threshold beyond which the inward-normal property no longer holds.Comment: 23 pages, 25 figures, Latex; full journal version with all proof
details. (Fixed crash-induced bugs in the abstract.
The Four Bars Problem
A four-bar linkage is a mechanism consisting of four rigid bars which are
joined by their endpoints in a polygonal chain and which can rotate freely at
the joints (or vertices). We assume that the linkage lies in the 2-dimensional
plane so that one of the bars is held horizontally fixed. In this paper we
consider the problem of reconfiguring a four-bar linkage using an operation
called a \emph{pop}. Given a polygonal cycle, a pop reflects a vertex across
the line defined by its two adjacent vertices along the polygonal chain. Our
main result shows that for certain conditions on the lengths of the bars of the
four-bar linkage, the neighborhood of any configuration that can be reached by
smooth motion can also be reached by pops. The proof relies on the fact that
pops are described by a map on the circle with an irrational number of
rotation.Comment: 18 page
The Theory of Bonds: A New Method for the Analysis of Linkages
In this paper we introduce a new technique, based on dual quaternions, for
the analysis of closed linkages with revolute joints: the theory of bonds. The
bond structure comprises a lot of information on closed revolute chains with a
one-parametric mobility. We demonstrate the usefulness of bond theory by giving
a new and transparent proof for the well-known classification of
overconstrained 5R linkages.Comment: more detailed explanations and additional reference
Elementary proofs of Kempe universality
An elementary proof is given to show that a parametrised algebraic curve in the plane may be traced out, in the sense of A. B. Kempe, by a finite pinned linkage. Additionally it is shown that any parametrised continuous curve \gamma:[0,1]\to \bR^2 may be traced out by an infinite linkage where the valencies of the joints is uniformly bounded. We also discuss related Kempe universality theorems and give a novel correction of Kempe's original argument
Detecting Weakly Simple Polygons
A closed curve in the plane is weakly simple if it is the limit (in the
Fr\'echet metric) of a sequence of simple closed curves. We describe an
algorithm to determine whether a closed walk of length n in a simple plane
graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time
algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary,
we obtain the first efficient algorithm to determine whether an arbitrary
n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We
also describe algorithms that detect weak simplicity in O(n log n) time for two
interesting classes of polygons. Finally, we discuss subtle errors in several
previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201
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