### Abstract

A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° 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 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.

Original language | English (US) |
---|---|

Pages (from-to) | 231-253 |

Number of pages | 23 |

Journal | Discrete and Computational Geometry |

Volume | 28 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2002 |

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### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

*Discrete and Computational Geometry*,

*28*(2), 231-253. https://doi.org/10.1007/s00454-002-2775-7

**Flipturning polygons.** / Aichholzer, Oswin; Cortés, Carmen; Demaine, Erik D.; Dujmović, Vida; Erickson, Jeff; Meijer, Henk; Overmars, Mark; Palop, Belén; Ramaswami, Suneeta; Toussaint, Godfried.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 28, no. 2, pp. 231-253. https://doi.org/10.1007/s00454-002-2775-7

}

TY - JOUR

T1 - Flipturning polygons

AU - Aichholzer, Oswin

AU - Cortés, Carmen

AU - Demaine, Erik D.

AU - Dujmović, Vida

AU - Erickson, Jeff

AU - Meijer, Henk

AU - Overmars, Mark

AU - Palop, Belén

AU - Ramaswami, Suneeta

AU - Toussaint, Godfried

PY - 2002/1/1

Y1 - 2002/1/1

N2 - A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° 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 n2-4n +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.

AB - A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° 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 n2-4n +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.

UR - http://www.scopus.com/inward/record.url?scp=0036025564&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036025564&partnerID=8YFLogxK

U2 - 10.1007/s00454-002-2775-7

DO - 10.1007/s00454-002-2775-7

M3 - Article

VL - 28

SP - 231

EP - 253

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 2

ER -