Part III: Polyhedra a: Folding Polygons Joseph ORourke Smith College.
Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.
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Transcript of Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.
![Page 1: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/1.jpg)
Part III: PolyhedraPart III: Polyhedrac: Cauchy’s Rigidity c: Cauchy’s Rigidity
TheoremTheorem
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
![Page 2: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/2.jpg)
Outline: Outline: Reconstruction of Convex Reconstruction of Convex PolyhedraPolyhedraCauchy to Sabitov (to an Open
Problem) Cauchy’s Rigidity Theorem Aleksandrov’s Theorem Sabitov’s Algorithm
![Page 3: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/3.jpg)
graphface anglesedge lengthsface areasface normalsdihedral anglesinscribed/circumscribed
Reconstruction of Convex Reconstruction of Convex PolyhedraPolyhedra
Steinitz’s Theorem
Minkowski’s Theorem}
![Page 4: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/4.jpg)
Minkowski’s TheoremMinkowski’s Theorem
![Page 5: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/5.jpg)
graphface anglesedge lengthsface areasface normalsdihedral anglesinscribed/circumscribed
Reconstruction of Convex Reconstruction of Convex PolyhedraPolyhedra
Cauchy’s Theorem
}
![Page 6: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/6.jpg)
Cauchy’s Rigidity TheoremCauchy’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
![Page 7: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/7.jpg)
Same facial structure,Same facial structure,noncongruent polyhedranoncongruent polyhedra
![Page 8: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/8.jpg)
Spherical polygonSpherical polygon
![Page 9: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/9.jpg)
Sign Labels: {+,-,0}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.
![Page 10: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/10.jpg)
The spherical polygon opens.
(a) Zero sign alternations; (b) Two sign alts.
![Page 11: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/11.jpg)
Sign changes Sign changes Euler Theorem Euler Theorem Contradiction Contradiction
Lemma ≥ 4 V
![Page 12: Part III: Polyhedra c: Cauchys Rigidity Theorem Joseph ORourke Smith College.](https://reader036.fdocuments.net/reader036/viewer/2022062618/5514a56a550346f06e8b5c4f/html5/thumbnails/12.jpg)
Flexing top of regular Flexing top of regular octahedronoctahedron