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Flipturning Polygons. (English) Zbl 1011.68146
Summary: A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity $$180^\circ$$ 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 $$\lfloor 5(n-4)/6 \rfloor$$ 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 prove that computing the longest flipturn sequence for a simple polygon is NP-hard. Finally, we show that although flipturn sequences for the same polygon can have significantly 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.

##### MSC:
 68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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