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Part I: Linkages a: Universality

Part I: Linkages a: Universality. Joseph O’Rourke Smith College. Outline. Chain Reachability Ruler Folding Pantograph Watt Linkage; Peaucellier Linkage Kempe Universality Theorem Kapovich & Millson Proof. Chain Reachability. Connectivity of configuration space Annulus

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Part I: Linkages a: Universality

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  1. Part I: Linkagesa: Universality Joseph O’Rourke Smith College

  2. Outline • Chain Reachability • Ruler Folding • Pantograph • Watt Linkage; Peaucellier Linkage • Kempe Universality Theorem • Kapovich & Millson Proof

  3. Chain Reachability • Connectivity of configuration space • Annulus • Two-kinks theorem

  4. Cinderella(FU Berlin, J.-R. Gebert & U. Kortenkamp) • Example construction • Lamp example [lamp1.cdy] • Cinderella 1.4: • http://page.inf.fu-berlin.de/~kortenka/CinderellaJapan/install.htm • user: cindybeta, password: geo-i.pdf • Cinderella 2: • http://www.cinderella.de/beta/install.htm • user: cindybeta, password: geo-i.pdf

  5. Steam Locomotive

  6. Watt Linkage • Circular to nearly linear • [Watt.cdy]

  7. Peaucellier Linkage • Circular to linear • [Peaucellier.cdy]

  8. Universality Theorems • Theorem 4.2.3 ([KM02]) Let C be a bounded portion of an algebraic curve in the plane. Then there exists a planar linkage such that the orbit of one joint is precisely C. • Theorem 4.2.4 ([JS99]) Let V ≤ Rd be a compact real algebraic variety (with topology induced by the Euclidean topology of Rd). Then V is homeomorphic to some components of a planar linkage.

  9. Translator

  10. Additor; Multiplicator • Additor • Multiplicator • [Contraparallelogram.cdy] • [Multiplicator.cdy]

  11. Additor

  12. Kempe Fig. 30

  13. Kempe Fig. 1

  14. Kempe Fig. 32

  15. Kempe Fig. 1

  16. Kempe: Parallelogram

  17. OverallConstruction

  18. Rhombus

  19. Kapovich & Millson 2002 • [Kem76] Alfred Bray Kempe. On a general method of describing plane curves of the nth degree by linkwork. Proc. London Math. Soc., 7:213-216, 1876. • [KM02] Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051-1107, 2002.

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