Abstract
Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( n2) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 219-236 |
| Number of pages | 18 |
| Journal | Computational Geometry: Theory and Applications |
| Volume | 31 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jun 1 2005 |
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Keywords
- Computational geometry
- Convexification
- Curve inflation
- Flips
- Flipturns
- Knot theory
- Molecular reconfiguration
- Pivots
- Polygonal linkages
- Polygons
- Polymer physics
- Robotics
- Self-avoiding walks
ASJC Scopus subject areas
- Geometry and Topology
- Computer Science Applications
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics
Cite this
The Erdös-Nagy theorem and its ramifications. / Toussaint, Godfried.
In: Computational Geometry: Theory and Applications, Vol. 31, No. 3, 01.06.2005, p. 219-236.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - The Erdös-Nagy theorem and its ramifications
AU - Toussaint, Godfried
PY - 2005/6/1
Y1 - 2005/6/1
N2 - Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( n2) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.
AB - Given a simple polygon in the plane, a flip is defined as follows: consider the convex hull of the polygon. If there are no pockets do not perform a flip. If there are pockets then reflect one pocket across its line of support of the polygon to obtain a new simple polygon. In 1934 Paul Erds introduced the problem of repeatedly flipping all the pockets of a simple polygon simultaneously and he conjectured that the polygon would become convex after a finite number of flips. In 1939 Béla Nagy proved that if at each step only one pocket is flipped the polygon will become convex after a finite number of flips. The history of this problem is reviewed, and a simple elementary proof is given of a stronger version of the theorem. Variants, generalizations, and applications of the theorem of interest in computational knot theory, polymer physics and molecular biology are discussed. Several results in the literature are improved with the application of the theorem. For example, Grünbaum and Zaks recently showed that even non-simple (self-crossing) polygons may be convexified in a finite number of suitable flips. Their flips each take Θ( n2) time to determine. A simpler proof of this result is given that yields an algorithm that takes O(n) time to determine each flip. In the context of knot theory Millet proposed an algorithm for convexifying equilateral polygons in 3-dimensions with a generalization of a flip called a pivot. Here Millet's algorithm is generalized so that it works also in dimensions higher than three and for polygons containing edges with arbitrary lengths. A list of open problems is included.
KW - Computational geometry
KW - Convexification
KW - Curve inflation
KW - Flips
KW - Flipturns
KW - Knot theory
KW - Molecular reconfiguration
KW - Pivots
KW - Polygonal linkages
KW - Polygons
KW - Polymer physics
KW - Robotics
KW - Self-avoiding walks
UR - http://www.scopus.com/inward/record.url?scp=84867967318&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84867967318&partnerID=8YFLogxK
U2 - 10.1016/j.comgeo.2004.12.005
DO - 10.1016/j.comgeo.2004.12.005
M3 - Article
AN - SCOPUS:84867967318
VL - 31
SP - 219
EP - 236
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
SN - 0925-7721
IS - 3
ER -