### Abstract

The classification of polygons is considered in which two polygons are regularly equivalent if one can be continuously transformed into the other such that for each intermediate no two adjacent edges overlap. A discrete analogue of the classic Whiney-Graustein theorem is proven by showing that the winding number of polygons is a complete invariant for this classification. Moreover, this proof is constructive in that for any pair of equivalent polygons, it produces some sequence of regular transformations taking one polygon to the other. Although this sequence has a quadratic number of transformations, it can be described and computed in real time.

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

Pages (from-to) | 603-621 |

Number of pages | 19 |

Journal | SIAM Journal on Computing |

Volume | 20 |

Issue number | 4 |

State | Published - Aug 1991 |

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### ASJC Scopus subject areas

- Computational Theory and Mathematics
- Applied Mathematics
- Theoretical Computer Science

### Cite this

*SIAM Journal on Computing*,

*20*(4), 603-621.

**Constructive Whitney-Graustein theorem. Or how to untangle closed planar curves.** / Mehlhorn, Kurt; Yap, Chee.

Research output: Contribution to journal › Article

*SIAM Journal on Computing*, vol. 20, no. 4, pp. 603-621.

}

TY - JOUR

T1 - Constructive Whitney-Graustein theorem. Or how to untangle closed planar curves

AU - Mehlhorn, Kurt

AU - Yap, Chee

PY - 1991/8

Y1 - 1991/8

N2 - The classification of polygons is considered in which two polygons are regularly equivalent if one can be continuously transformed into the other such that for each intermediate no two adjacent edges overlap. A discrete analogue of the classic Whiney-Graustein theorem is proven by showing that the winding number of polygons is a complete invariant for this classification. Moreover, this proof is constructive in that for any pair of equivalent polygons, it produces some sequence of regular transformations taking one polygon to the other. Although this sequence has a quadratic number of transformations, it can be described and computed in real time.

AB - The classification of polygons is considered in which two polygons are regularly equivalent if one can be continuously transformed into the other such that for each intermediate no two adjacent edges overlap. A discrete analogue of the classic Whiney-Graustein theorem is proven by showing that the winding number of polygons is a complete invariant for this classification. Moreover, this proof is constructive in that for any pair of equivalent polygons, it produces some sequence of regular transformations taking one polygon to the other. Although this sequence has a quadratic number of transformations, it can be described and computed in real time.

UR - http://www.scopus.com/inward/record.url?scp=0026206775&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026206775&partnerID=8YFLogxK

M3 - Article

VL - 20

SP - 603

EP - 621

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

SN - 0097-5397

IS - 4

ER -