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Part I: Linkages a: Universality Joseph ORourke Smith College.

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Presentation on theme: "Part I: Linkages a: Universality Joseph ORourke Smith College."— Presentation transcript:

1 Part I: Linkages a: Universality Joseph ORourke Smith College

2 Outline zChain Reachability zRuler Folding zPantograph zWatt Linkage; Peaucellier Linkage zKempe Universality Theorem zKapovich & Millson Proof

3 Chain Reachability zConnectivity of configuration space zAnnulus zTwo-kinks theorem

4 Cinderella Cinderella (FU Berlin, J.-R. Gebert & U. Kortenkamp) Cinderella zExample construction zLamp example [lamp1.cdy]lamp1.cdy z Cinderella 1.4: yhttp://page.inf.fu-berlin.de/~kortenka/CinderellaJapan/install.htm yuser: cindybeta, password: geo-i.pdf zCinderella 2: yhttp://www.cinderella.de/beta/install.htm yuser: cindybeta, password: geo-i.pdf

5 Steam Locomotive

6 Watt Linkage zCircular to nearly linear z[Watt.cdy]Watt.cdy

7 Peaucellier Linkage zCircular to linear z[Peaucellier.cdy]Peaucellier.cdy

8 Universality Theorems zTheorem ([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. zTheorem ([JS99]) Let V R d be a compact real algebraic variety (with topology induced by the Euclidean topology of R d ). Then V is homeomorphic to some components of a planar linkage.

9 Translator

10 Additor; Multiplicator zAdditor zMultiplicator z[Contraparallelogram.cdy]Contraparallelogram.cdy z[Multiplicator.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 Overall Construction

18 Rhombus

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


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