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

Kurt Mehlhorn, Chee Keng Yap

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)603-621
Number of pages19
JournalSIAM Journal on Computing
Issue number4
StatePublished - 1991

ASJC Scopus subject areas

  • General Computer Science
  • General Mathematics


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