Part I: Linkages c: Locked Chains
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Transcript of Part I: Linkages c: Locked Chains
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Part I: LinkagesPart I: Linkagesc: Locked Chainsc: Locked Chains
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
(Many slides made by Erik (Many slides made by Erik Demaine)Demaine)
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OutlineOutline
Locked Chains in 3DLocked Trees in 2DNo Locked Chains in 2DAlgorithms for Unlocking Chains in 2D
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Linkages / FrameworksLinkages / Frameworks
Bar / link / edge = line segmentVertex / joint = connection between
endpoints of bars
Closed chain / cycle / polygon
Open chain / arc Tree General
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ConfigurationsConfigurations
Configuration = positions of the vertices that preserves the bar lengths
Non-self-intersecting configurations Self-intersecting
Non-self-intersecting = No bars cross
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Locked QuestionLocked Question
Can a linkage be moved between any twonon-self-intersecting configurations?
?
Can any non-self-intersecting configuration be unfolded, i.e., moved to “canonical” configuration? Equivalent by reversing and concatenating motions
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Canonical ConfigurationsCanonical Configurations
Arcs: Straight configuration
Cycles: Convex configurations
Trees: Flat configurations
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What Linkages Can Lock?What Linkages Can Lock?[Schanuel & Bergman, early 1970’s; Grenander [Schanuel & Bergman, early 1970’s; Grenander 1987; Lenhart & Whitesides 1991; Mitchell 1992]1987; Lenhart & Whitesides 1991; Mitchell 1992]Can every chain be straightened?Can every cycle be convexified?Can every tree be flattened?
Chains Cycles Trees
2D Yes Yes No3D No No No4D & higher Yes Yes Yes
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Locked 3D Chains Locked 3D Chains [Cantarella & [Cantarella & Johnston 1998; Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999]Whitesides 1999]Cannot straighten some chains
Idea of proof: Ends must be far away from the turns Turns must stay relatively close to each other Could effectively connect ends together Hence, any straightening unties a trefoil knot
Sphere separates turns from ends
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Locked 3D Chains Locked 3D Chains [Cantarella & [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999]Toussaint, Whitesides 1999]Double this chain:
This unknotted cycle cannot be convexified by the same argument
Several locked hexagons are also known
Cantarella & Johnston 1998
Toussaint 1999
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Locked 2D TreesLocked 2D Trees[Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, [Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Robbins, Streinu, Toussaint, Whitesides 1998]Robbins, Streinu, Toussaint, Whitesides 1998]
Theorem: Not all trees can be flattened No petal can be opened unless all others
are closed significantly No petal can be closed more than a little
unless it has already opened
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Converting the Tree into a Converting the Tree into a CycleCycleDouble each edge:
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Converting the Tree into a Converting the Tree into a CycleCycleBut this cycle can be convexified:
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Converting the Tree into a Converting the Tree into a CycleCycleBut this cycle can be convexified:
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One Key Idea for 2D Cycles:One Key Idea for 2D Cycles:Increasing DistancesIncreasing DistancesA motion is expansive if no inter-
vertex distances decreasesLemma: If a motion is expansive,
the framework cannot cross itself
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TheoremTheorem[Connelly, Demaine, Rote 2000][Connelly, Demaine, Rote 2000]For any family of chains and cycles,
there is a motion that Makes the chains straight Makes the cycles convex Increases most pairwise distances (and area)
Except: Chains or cycles contained within a cycle might not be straightened or convexified
Furthermore:Motion preserves symmetries andis piecewise-differentiable (smooth)
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Algorithms for 2D ChainsAlgorithms for 2D Chains
Connelly, Demaine, Rote (2000) — ODE + convex programming
Streinu (2000) — pseudotriangulations + piecewise-algebraic motions
Cantarella, Demaine, Iben, O’Brien (2003) — energy
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Energy AlgorithmEnergy Algorithm[Cantarella, Demaine, Iben, O’Brien][Cantarella, Demaine, Iben, O’Brien]Use ideas from knot energies
to evolve a linkage via gradient descentLoosen expansiveness constraint;
still avoid crossingsResulting motion is simpler
C (instead of piecewise-C1 or piecewise-C) Easy to compute, even physically In polynomial time, produce simplest possible
explicit representation: piecewise-linear Preserves initial symmetries in the linkage
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Basic IdeaBasic Idea
Define energy function on configurationsso that Crossing requires infinite energy Expansive motions decrease energy Minimum-energy configuration
is straight/convexFollow any energy-decreasing motion
Guaranteed to exist by expansive motion Not necessarily expansive, but avoids
crossings Smooth (C) motion preserving symmetries
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Euclidean-Distance EnergyEuclidean-Distance Energy
C1,1 (Lipschitz)Charge ( @ boundary)Repulsive (expansive)Separable (components)
.,,
2),(1
evEeVv evd
Energy field applied toan additional point noton the white chain,ignoring nearest terms
ev
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Visual ComparisonVisual Comparison
CDR
Energy
CDR
Energy
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Energy ExamplesEnergy Examples
spiral
spider
tentacle
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Energy AnimationsEnergy Animations http://www.cs.berkeley.edu/b-cam/Papers/Cantarella
-2004-AED/index.htmlteeth.avitree.avidoubleSpiral.avispider.avitentacle.avi