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Part III: Polyhedra c: Cauchy’s Rigidity Theorem. Joseph O’Rourke Smith College. Outline: Reconstruction of Convex Polyhedra. Cauchy to Sabitov (to an Open Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm. Reconstruction of Convex Polyhedra. graph

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Part iii polyhedra c cauchy s rigidity theorem l.jpg

Part III: Polyhedrac: Cauchy’s Rigidity Theorem

Joseph O’Rourke

Smith College


Outline reconstruction of convex polyhedra l.jpg
Outline: Reconstruction of Convex Polyhedra

  • Cauchy to Sabitov (to an Open Problem)

    • Cauchy’s Rigidity Theorem

    • Aleksandrov’s Theorem

    • Sabitov’s Algorithm


Reconstruction of convex polyhedra l.jpg
Reconstruction of Convex Polyhedra

  • graph

  • face angles

  • edge lengths

  • face areas

  • face normals

  • dihedral angles

  • inscribed/circumscribed

Steinitz’s Theorem

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Minkowski’s Theorem



Reconstruction of convex polyhedra5 l.jpg
Reconstruction of Convex Polyhedra

  • graph

  • face angles

  • edge lengths

  • face areas

  • face normals

  • dihedral angles

  • inscribed/circumscribed

}

Cauchy’s Theorem


Cauchy s rigidity theorem l.jpg
Cauchy’s Rigidity Theorem

  • If two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent;

  • in particular, the dihedral angles at each edge are the same.

Global rigidity == unique realization


Same facial structure noncongruent polyhedra l.jpg
Same facial structure,noncongruent polyhedra



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Sign Labels: {+,-,0}

  • Compare spherical polygons Q to Q’

  • Mark vertices according to dihedral angles: {+,-,0}.

Lemma: The total number of alternations in sign around the boundary of Q is ≥ 4.


The spherical polygon opens l.jpg
The spherical polygon opens.

(a) Zero sign alternations; (b) Two sign alts.


Sign changes euler theorem contradiction l.jpg
Sign changes  Euler Theorem Contradiction

Lemma  ≥ 4 V