Part I: Linkages c: Locked Chains Joseph ORourke Smith College (Many slides made by Erik Demaine)

Outline zLocked Chains in 3D zLocked Trees in 2D zNo Locked Chains in 2D zAlgorithms for Unlocking Chains in 2D

Linkages / Frameworks zBar / link / edge = line segment zVertex / joint = connection between endpoints of bars Closed chain / cycle / polygon Open chain / arc TreeGeneral

Configurations zConfiguration = positions of the vertices that preserves the bar lengths Non-self-intersecting configurations Self-intersecting zNon-self-intersecting = No bars cross

Locked Question zCan a linkage be moved between any two non-self-intersecting configurations? ? zCan any non-self-intersecting configuration be unfolded, i.e., moved to canonical configuration? yEquivalent by reversing and concatenating motions

Canonical Configurations zArcs: Straight configuration zCycles: Convex configurations zTrees: Flat configurations

What Linkages Can Lock? [Schanuel & Bergman, early 1970s; Grenander 1987; Lenhart & Whitesides 1991; Mitchell 1992] zCan every chain be straightened? zCan every cycle be convexified? zCan every tree be flattened? ChainsCyclesTrees 2DYes No 3DNo 4D & higher Yes

Locked 3D Chains [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999] zCannot straighten some chains zIdea of proof: yEnds must be far away from the turns yTurns must stay relatively close to each other y Could effectively connect ends together yHence, any straightening unties a trefoil knot Sphere separates turns from ends

Locked 3D Chains [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999] zDouble this chain: zThis unknotted cycle cannot be convexified by the same argument zSeveral locked hexagons are also known Cantarella & Johnston 1998 Toussaint 1999

Locked 2D Trees [Biedl, Demaine, Demaine, Lazard, Lubiw, ORourke, Robbins, Streinu, Toussaint, Whitesides 1998] zTheorem: Not all trees can be flattened yNo petal can be opened unless all others are closed significantly yNo petal can be closed more than a little unless it has already opened

Converting the Tree into a Cycle zDouble each edge:

Converting the Tree into a Cycle zBut this cycle can be convexified:

Converting the Tree into a Cycle zBut this cycle can be convexified:

One Key Idea for 2D Cycles: Increasing Distances zA motion is expansive if no inter-vertex distances decreases zLemma: If a motion is expansive, the framework cannot cross itself

Theorem [Connelly, Demaine, Rote 2000] zFor any family of chains and cycles, there is a motion that yMakes the chains straight yMakes the cycles convex yIncreases most pairwise distances (and area) zExcept: Chains or cycles contained within a cycle might not be straightened or convexified zFurthermore: Motion preserves symmetries and is piecewise-differentiable (smooth)

Conclusion zConstructive proof that every polygonal chain can be straightened and every polygon can be convexified yBased on flow through a vector field defined implicitly by an optimization problem yNot technically a finite algorithm yEasy to approximate in practice yConsequences: xPiecewise-differentiable xPreserve symmetries of linkage xConfiguration space is contractible

Algorithms for 2D Chains Connelly, Demaine, Rote (2000) ODE + convex programming Streinu (2000) pseudotriangulations + piecewise-algebraic motions Cantarella, Demaine, Iben, OBrien (2003) energy

Energy Algorithm [Cantarella, Demaine, Iben, OBrien] zUse ideas from knot energies to evolve a linkage via gradient descent zLoosen expansiveness constraint; still avoid crossings zResulting motion is simpler yC (instead of piecewise-C 1 or piecewise-C ) yEasy to compute, even physically yIn polynomial time, produce simplest possible explicit representation: piecewise-linear yPreserves initial symmetries in the linkage

Basic Idea zDefine energy function on configurations so that yCrossing requires infinite energy yExpansive motions decrease energy yMinimum-energy configuration is straight/convex zFollow any energy-decreasing motion yGuaranteed to exist by expansive motion yNot necessarily expansive, but avoids crossings ySmooth (C ) motion preserving symmetries

Euclidean-Distance Energy zC 1,1 (Lipschitz) zCharge boundary) zRepulsive (expansive) zSeparable (components) Energy field applied to an additional point not on the white chain, ignoring nearest terms e v

Algorithm zParameterize to keep bars fixed length and cycles closed y(Cosines of) angles, except for some in cycles yCompute location of final vertex in cycle by intersecting two circles zCompute Euclidean gradient in O(n 2 ) time zFollow gradient linearly by a magnitude that decreases energy best zUse bounds on convergence of steepest descent θ1θ1 θ2θ2 θ3θ3 θ4θ4 θ5θ5 θ6θ6 θ7θ7 θ8θ8 θ9θ9

Visual Comparison CDR Energy CDR Energy

Energy Examples spiral spider tentacle

Energy Animations zhttp:// AED/index.htmlhttp:// AED/index.html zteeth.avi ztree.avi zdoubleSpiral.avi zspider.avi ztentacle.avi