Flipturning polygons

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

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)231-253
Number of pages23
JournalDiscrete and Computational Geometry
Volume28
Issue number2
DOIs
StatePublished - Jan 1 2002

Fingerprint

Simple Polygon
Polygon
Convex polygon
Orthogonal Polygons
Transform
Midpoint
Concavity
Convex Hull
Rotating
NP-complete problem
Upper bound
Computing
Arbitrary

ASJC Scopus subject areas

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

Cite this

Aichholzer, O., Cortés, C., Demaine, E. D., Dujmović, V., Erickson, J., Meijer, H., ... Toussaint, G. (2002). Flipturning polygons. 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.

In: Discrete and Computational Geometry, Vol. 28, No. 2, 01.01.2002, p. 231-253.

Research output: Contribution to journalArticle

Aichholzer, O, Cortés, C, Demaine, ED, Dujmović, V, Erickson, J, Meijer, H, Overmars, M, Palop, B, Ramaswami, S & Toussaint, G 2002, 'Flipturning polygons', Discrete and Computational Geometry, vol. 28, no. 2, pp. 231-253. https://doi.org/10.1007/s00454-002-2775-7
Aichholzer O, Cortés C, Demaine ED, Dujmović V, Erickson J, Meijer H et al. Flipturning polygons. Discrete and Computational Geometry. 2002 Jan 1;28(2):231-253. https://doi.org/10.1007/s00454-002-2775-7
Aichholzer, Oswin ; Cortés, Carmen ; Demaine, Erik D. ; Dujmović, Vida ; Erickson, Jeff ; Meijer, Henk ; Overmars, Mark ; Palop, Belén ; Ramaswami, Suneeta ; Toussaint, Godfried. / Flipturning polygons. In: Discrete and Computational Geometry. 2002 ; Vol. 28, No. 2. pp. 231-253.
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