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Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College

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Outline: Reconstruction of Convex Polyhedra zCauchy to Sabitov (to an Open Problem) yCauchys Rigidity Theorem yAleksandrovs Theorem ySabitovs Algorithm

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zgraph zface angles zedge lengths zface areas zface normals zdihedral angles zinscribed/circumscribed Reconstruction of Convex Polyhedra Steinitzs Theorem Minkowskis Theorem }

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zgraph zface angles zedge lengths zface areas zface normals zdihedral angles zinscribed/circumscribed Reconstruction of Convex Polyhedra Cauchys Theorem }

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Cauchys Rigidity Theorem zIf two closed, convex polyhedra are combinatorially equivalent, with corresponding faces congruent, then the polyhedra are congruent; zin particular, the dihedral angles at each edge are the same. Global rigidity == unique realization

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Same facial structure, noncongruent polyhedra

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Spherical polygon

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Sign Labels: {+,-,0} zCompare spherical polygons Q to Q zMark vertices according to dihedral angles: {+,-,0}. Lemma: The total number of alternations in sign around the boundary of Q is 4.

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The spherical polygon opens. (a) Zero sign alternations; (b) Two sign alts.

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Sign changes Euler Theorem Contradiction Lemma 4 V

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Flexing top of regular octahedron

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