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

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

Part I: Linkages a: Universality Joseph ORourke Smith College

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

Chain Reachability zConnectivity of configuration space zAnnulus zTwo-kinks theorem

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

Steam Locomotive

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

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

Universality Theorems zTheorem 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. zTheorem 4.2.4 ([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.

Translator

Kempe Fig. 30

Kempe Fig. 1

Kempe Fig. 32

Kempe Fig. 1

Kempe: Parallelogram

Overall Construction

Rhombus

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., 7:213-216, 1876. z[KM02] Michael Kapovich and John J. Millson. Universality theorems for configuration spaces of planar linkages. Topology, 41(6):1051-1107, 2002.