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Flat Foldings of Plane Graphs with Prescribed Angles and Edge Lengths

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 نشر من قبل David Eppstein
 تاريخ النشر 2014
  مجال البحث الهندسة المعلوماتية
والبحث باللغة English




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When can a plane graph with prescribed edge lengths and prescribed angles (from among ${0,180^circ, 360^circ$}) be folded flat to lie in an infinitesimally thin line, without crossings? This problem generalizes the classic theory of single-vertex flat origami with prescribed mountain-valley assignment, which corresponds to the case of a cycle graph. We characterize such flat-foldable plane graphs by two obviously necessary but also sufficient conditions, proving a conjecture made in 2001: the angles at each vertex should sum to $360^circ$, and every face of the graph must itself be flat foldable. This characterization leads to a linear-time algorithm for testing flat foldability of plane graphs with prescribed edge lengths and angles, and a polynomial-time algorithm for counting the number of distinct folded states.



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