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