Part I: Linkages a: Universality Joseph ORourke Smith College.
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Transcript of Part I: Linkages a: Universality Joseph ORourke Smith College.
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Part I: LinkagesPart I: Linkagesa: Universalitya: Universality
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
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OutlineOutline
Chain ReachabilityRuler FoldingPantographWatt Linkage; Peaucellier LinkageKempe Universality TheoremKapovich & Millson Proof
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Chain ReachabilityChain Reachability
Connectivity of configuration spaceAnnulusTwo-kinks theorem
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CinderellaCinderella (FU Berlin, J.-R. Gebert & U. (FU Berlin, J.-R. Gebert & U. Kortenkamp)Kortenkamp)
Example constructionLamp 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
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Steam LocomotiveSteam Locomotive
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Watt LinkageWatt Linkage
Circular to nearly linear
[Watt.cdy]
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Peaucellier LinkagePeaucellier Linkage
Circular to linear[Peaucellier.cdy]
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Universality TheoremsUniversality 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.
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TranslatorTranslator
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Additor; MultiplicatorAdditor; Multiplicator
AdditorMultiplicator[Contraparallelogram.cdy][Multiplicator.cdy]
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AdditorAdditor
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Kempe Fig. 30Kempe Fig. 30
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Kempe Fig. 1Kempe Fig. 1
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Kempe Fig. 32Kempe Fig. 32
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Kempe Fig. 1Kempe Fig. 1
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Kempe: ParallelogramKempe: Parallelogram
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OverallOverallConstructionConstruction
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RhombusRhombus
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Kapovich & Millson 2002Kapovich & 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.