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Linkages and Universality: From Chain Reachability to Planar Linkages

This article explores the concepts of chain reachability and planar linkages, showcasing examples such as the Watt Linkage and Peaucellier Linkage. It also delves into the Kempe Universality Theorem and the Kapovich & Millson Proof. The Cinderella software and steam locomotive applications are discussed as well.

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Linkages and Universality: From Chain Reachability to Planar Linkages

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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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