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Part I: Linkages c: Locked Chains

This presentation uncovers the complexity of locked chains and trees in linkages, exploring the configurations in 2D and 3D space. Investigate questions of unlocking chains and canonical configurations, delve into locked structures in both 2D and 3D realms, and analyze the transformations of trees into cycles. Discover key concepts like expansiveness in motions, shape transformations, and energy algorithms for evolving linkages. Explore the visual representations and animations to enhance your understanding of these intricate linkages and frameworks.

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Part I: Linkages c: Locked Chains

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  1. Part I: Linkagesc: Locked Chains Joseph O’Rourke Smith College (Many slides made by Erik Demaine)

  2. Outline • Locked Chains in 3D • Locked Trees in 2D • No Locked Chains in 2D • Algorithms for Unlocking Chains in 2D

  3. Linkages / Frameworks • Bar / link / edge = line segment • Vertex / joint = connection between endpoints of bars Closed chain / cycle / polygon Open chain / arc Tree General

  4. Configurations • Configuration = positions of the vertices that preserves the bar lengths • Non-self-intersecting = No bars cross Self-intersecting Non-self-intersecting configurations

  5. Locked 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 ?

  6. Canonical Configurations • Arcs: Straightconfiguration • Cycles: Convexconfigurations • Trees: Flat configurations

  7. What Linkages Can Lock?[Schanuel & Bergman, early 1970’s; Grenander 1987; Lenhart & Whitesides 1991; Mitchell 1992] • Can every chain be straightened? • Can every cycle be convexified? • Can every tree be flattened?

  8. Locked 3D Chains [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, 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

  9. Toussaint 1999 Cantarella & Johnston 1998 Locked 3D Chains [Cantarella & Johnston 1998; Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, Overmars, Robbins, Streinu, Toussaint, Whitesides 1999] • Double this chain: • This unknotted cycle cannot be convexified by the same argument • Several locked hexagons are also known

  10. Locked 2D Trees[Biedl, Demaine, Demaine, Lazard, Lubiw, O’Rourke, 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

  11. Converting the Tree into a Cycle • Double each edge:

  12. Converting the Tree into a Cycle • But this cycle can be convexified:

  13. Converting the Tree into a Cycle • But this cycle can be convexified:

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

  15. Theorem[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)

  16. Algorithms for 2D Chains Connelly, Demaine, Rote (2000) — ODE + convex programming Streinu (2000) — pseudotriangulations + piecewise-algebraic motions Cantarella, Demaine, Iben, O’Brien (2003) — energy

  17. Energy Algorithm[Cantarella, Demaine, Iben, O’Brien] • Use ideas from knot energiesto evolve a linkage via gradient descent • Loosen expansiveness constraint;still avoid crossings • Resulting 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

  18. Basic Idea • Define energy function on configurationsso that • Crossing requires infinite energy • Expansive motions decrease energy • Minimum-energy configurationis straight/convex • Follow any energy-decreasing motion • Guaranteed to exist by expansive motion • Not necessarily expansive, but avoids crossings • Smooth (C) motion preserving symmetries

  19. Euclidean-Distance Energy • C1,1(Lipschitz) • Charge ( @ boundary) • Repulsive (expansive) • Separable (components) e v Energy field applied toan additional point noton the white chain,ignoring nearest terms

  20. CDR Energy CDR Energy Visual Comparison

  21. Energy Examples spiral spider tentacle

  22. Energy Animations • http://www.cs.berkeley.edu/b-cam/Papers/Cantarella-2004-AED/index.html • teeth.avi • tree.avi • doubleSpiral.avi • spider.avi • tentacle.avi

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