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Straightening polygonal arcs and convexifying polygonal cycles. (English) Zbl 1046.52016
A planar polygonal arc or open polygonal chain is a sequence of finitely many line segments in the plane connected in a path without self-intersections. It has been an outstanding question as to whether it is possible to move a polygonal arc continuously in such a way that each edge remains a fixed length, there are no self-intersections during the motion, and at the end of the motion the arc lies on a straight line. This has come to be known as the carpenter’s rule problem. A related question is whether it is possible to move a polygonal simple closed curve continuously in the plane, often called a closed polygonal chain or polygon, again without creating self-intersections or changing the lengths of the edges, so that it ends up a convex closed curve.
The authors solve both problems by showing that in both cases there is such a motion.

MSC:
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
51E12 Generalized quadrangles and generalized polygons in finite geometry
52B55 Computational aspects related to convexity
55U15 Chain complexes in algebraic topology
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