Part I: LinkagesPart I: Linkagesa: Universalitya: Universality

Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College

OutlineOutline

Chain ReachabilityRuler FoldingPantographWatt Linkage; Peaucellier LinkageKempe Universality TheoremKapovich & Millson Proof

Chain ReachabilityChain Reachability

Connectivity of configuration spaceAnnulusTwo-kinks theorem

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

Steam LocomotiveSteam Locomotive

Watt LinkageWatt Linkage

Circular to nearly linear

[Watt.cdy]

Peaucellier LinkagePeaucellier Linkage

Circular to linear[Peaucellier.cdy]

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.

TranslatorTranslator

Additor; MultiplicatorAdditor; Multiplicator

AdditorMultiplicator[Contraparallelogram.cdy][Multiplicator.cdy]

AdditorAdditor

Kempe Fig. 30Kempe Fig. 30

Kempe Fig. 1Kempe Fig. 1

Kempe Fig. 32Kempe Fig. 32

Kempe Fig. 1Kempe Fig. 1

Kempe: ParallelogramKempe: Parallelogram

OverallOverallConstructionConstruction

RhombusRhombus

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.