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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: Polyhedrac: 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

• graph

• face angles

• edge lengths

• face areas

• face normals

• dihedral angles

• inscribed/circumscribed

Steinitz’s Theorem

}

Minkowski’s Theorem

• graph

• face angles

• edge lengths

• face areas

• face normals

• dihedral angles

• inscribed/circumscribed

}

Cauchy’s 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

• 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.

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

Sign changes  Euler Theorem Contradiction

Lemma  ≥ 4 V