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Part I: Linkages a: Universality Joseph ORourke Smith College

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

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Chain Reachability zConnectivity of configuration space zAnnulus zTwo-kinks theorem

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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

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Steam Locomotive

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

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

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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.

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Translator

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Additor; Multiplicator zAdditor zMultiplicator z[Contraparallelogram.cdy]Contraparallelogram.cdy z[Multiplicator.cdy]Multiplicator.cdy

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Additor

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Kempe Fig. 30

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Kempe Fig. 1

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Kempe Fig. 32

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Kempe Fig. 1

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Kempe: Parallelogram

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Overall Construction

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Rhombus

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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.

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Instructions for Lesson Delivery 1.This is lesson 1 of 2 for Linear equations. 2.Print the last page for all students so they can place them in their whiteboards.

Instructions for Lesson Delivery 1.This is lesson 1 of 2 for Linear equations. 2.Print the last page for all students so they can place them in their whiteboards.

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