33 research outputs found
Convexity-Increasing Morphs of Planar Graphs
We study the problem of convexifying drawings of planar graphs. Given any
planar straight-line drawing of an internally 3-connected graph, we show how to
morph the drawing to one with strictly convex faces while maintaining planarity
at all times. Our morph is convexity-increasing, meaning that once an angle is
convex, it remains convex. We give an efficient algorithm that constructs such
a morph as a composition of a linear number of steps where each step either
moves vertices along horizontal lines or moves vertices along vertical lines.
Moreover, we show that a linear number of steps is worst-case optimal.
To obtain our result, we use a well-known technique by Hong and Nagamochi for
finding redrawings with convex faces while preserving y-coordinates. Using a
variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and
Nagamochi's result which comes with a better running time. This is of
independent interest, as Hong and Nagamochi's technique serves as a building
block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201
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
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.
Deformations of associahedra and visibility graphs
Given an arbitrary simple polygon, we construct a polytopal complex analogous to the associahedron based on its convex diagonalizations. This polytopal complex is shown to be contractible, and a geometric realization is provided based on the theory of secondary polytopes. We then reformulate a combinatorial deformation theory in terms of visibility and presents some open problems
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.
Pseudo-Triangulations, Rigidity and Motion Planning
This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc
Flipturning polygons
A flipturn is an operation that transforms a nonconvex simple polygon into
another simple polygon, by rotating a concavity 180 degrees around the midpoint
of its bounding convex hull edge. Joss and Shannon proved in 1973 that a
sequence of flipturns eventually transforms any simple polygon into a convex
polygon. This paper describes several new results about such flipturn
sequences. We show that any orthogonal polygon is convexified after at most n-5
arbitrary flipturns, or at most 5(n-4)/6 well-chosen flipturns, improving the
previously best upper bound of (n-1)!/2. We also show that any simple polygon
can be convexified by at most n^2-4n+1 flipturns, generalizing earlier results
of Ahn et al. These bounds depend critically on how degenerate cases are
handled; we carefully explore several possibilities. We describe how to
maintain both a simple polygon and its convex hull in O(log^4 n) time per
flipturn, using a data structure of size O(n). We show that although flipturn
sequences for the same polygon can have very different lengths, the shape and
position of the final convex polygon is the same for all sequences and can be
computed in O(n log n) time. Finally, we demonstrate that finding the longest
convexifying flipturn sequence of a simple polygon is NP-hard.Comment: 26 pages, 32 figures, see also
http://www.uiuc.edu/~jeffe/pubs/flipturn.htm