[Paper Review] Flipturning polygons

Oswin Aichholzer, Carmen Cortés+8 more|ArXiv.org|Aug 16, 2000

Computational Geometry and Mesh Generation4 citations

TL;DR

This paper investigates flipturn operations—180° rotations of polygon concavities (pockets) about the midpoint of their bounding edges—to convexify simple polygons. It establishes tight bounds on flipturn sequences: orthogonal polygons are convexified in at most $n-5$ arbitrary flipturns or $\lfloor 5(n-4)/6\rfloor$ well-chosen ones, and general simple polygons in at most $n^2 - 4n + 1$ flipturns. The final convex shape is unique across all sequences and computable in $O(n\log n)$ time.

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.

Motivation & Objective

• To establish tighter upper bounds on the number of flipturns required to convexify simple and orthogonal polygons.
• To analyze the uniqueness of the final convex polygon across all flipturn sequences.
• To investigate the computational complexity of finding the longest flipturn sequence for convexification.
• To design an efficient data structure for maintaining a polygon and its convex hull during flipturn operations.

Proposed method

• Define a flipturn as rotating a polygon's pocket 180° around the midpoint of its lid, reversing edge order without changing lengths or orientations.
• Use combinatorial and geometric analysis to bound flipturn sequences for orthogonal and general simple polygons.
• Construct a reduction from the Subset Sum problem to prove NP-hardness of finding the longest flipturn sequence.
• Design a data structure that maintains the polygon and its convex hull in $O(\log^4 n)$ time per flipturn using $O(n)$ space.
• Prove that the final convex polygon is independent of the flipturn sequence, enabling $O(n\log n)$ computation of the canonical convex shape.
• Analyze degeneracy cases carefully to ensure bounds are tight and robust under various polygon configurations.

Experimental results

Research questions

• RQ1 What is the maximum number of flipturns required to convexify any simple polygon, and how does this depend on polygon type?
• RQ2 Can the final convex shape produced by any flipturn sequence be computed efficiently and uniquely?
• RQ3 Is finding the longest flipturn sequence required to convexify a polygon computationally hard?
• RQ4 How efficiently can a polygon and its convex hull be maintained during dynamic flipturn operations?
• RQ5 Do orthogonal polygons require fewer flipturns than general simple polygons, and what are the tightest known bounds?

Key findings

• Any orthogonal polygon can be convexified in at most $n - 5$ arbitrary flipturns, improving the prior bound of $(n-1)!/2$.
• For orthogonal polygons, at most $\lfloor 5(n-4)/6\rfloor$ well-chosen flipturns are needed to achieve convexification.
• General simple polygons can be convexified in at most $n^2 - 4n + 1$ flipturns, generalizing earlier results.
• The final convex polygon is identical for all flipturn sequences, and can be computed in $O(n\log n)$ time.
• Maintaining a simple polygon and its convex hull during flipturns is possible in $O(\log^4 n)$ time per operation with $O(n)$ space.
• Finding the longest flipturn sequence that convexifies a given simple polygon is NP-hard, even for orthogonal polygons.