Convexifying polygons with simple projections
Information Processing Letters
Interlocked open linkages with few joints
Proceedings of the eighteenth annual symposium on Computational geometry
Polygonal chains cannot lock in 4D
Computational Geometry: Theory and Applications
Proceedings of the nineteenth annual symposium on Computational geometry
A combinatorial approach to planar non-colliding robot arm motion planning
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
An energy-driven approach to linkage unfolding
SCG '04 Proceedings of the twentieth annual symposium on Computational geometry
On straightening low-diameter unit trees
GD'05 Proceedings of the 13th international conference on Graph Drawing
On Unfolding 3D Lattice Polygons and 2D Orthogonal Trees
COCOON '08 Proceedings of the 14th annual international conference on Computing and Combinatorics
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
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We consider the problems of straightening polygonal trees and convexifying polygons by continuous motions such that rigid edges can rotate around vertex joints and no edge crossings are allowed. A tree can be straightened if all its edges can be aligned along a common straight line such that each edge points “away” from a designated leaf node. A polygon can be convexified if it can be reconfigured to a convex polygon. A lattice tree (resp. polygon) is a tree (resp. polygon) containing only edges from a square or cubic lattice. We first show that a 2D lattice chain or a 3D lattice tree can be straightened efficiently in O(n) moves and time, where n is the number of tree edges. We then show that a 2D lattice tree can be straightened efficiently in O(n2) moves and time. Furthermore, we prove that a 2D lattice polygon or a 3D lattice polygon with simple shadow can be convexified efficiently in O(n2) moves and time. Finally, we show that two special classes of diameter-4 trees in two dimensions can always be straightened.