Flipturning polygons
Abstract
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 n5 arbitrary flipturns, or at most 5(n4)/6 wellchosen flipturns, improving the previously best upper bound of (n1)!/2. We also show that any simple polygon can be convexified by at most n^24n+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 NPhard.
 Publication:

arXiv eprints
 Pub Date:
 August 2000
 arXiv:
 arXiv:cs/0008010
 Bibcode:
 2000cs........8010A
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Discrete Mathematics;
 Mathematics  Metric Geometry;
 F.2.2;
 G.2
 EPrint:
 26 pages, 32 figures, see also http://www.uiuc.edu/~jeffe/pubs/flipturn.html