Flipturning Polygons

O. Aichholzer, C. Cortés, E. Demaine, V. Dujmovic, J. Erickson, H. Meijer, M. Overmars, B. Palop, S. Ramaswami, and G. Toussaint

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 $n-5$ 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.

Reference: O. Aichholzer, C. Cortés, E. Demaine, V. Dujmovic, J. Erickson, H. Meijer, M. Overmars, B. Palop, S. Ramaswami, and G. Toussaint. Flipturning polygons. In Proc. Japan Conference on Discrete and Computational Geometry JCDCG 2000, Tokay University, Tokyo, Japan, 2000.