6.849: Geometric Folding Algorithms: Linkages, Origami, Polyhedra (Fall 2012)

Prof. Erik Demaine       TA: Jayson Lynch

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[+] Locked linkages: Why expansiveness, energy algorithm correctness, pointed pseudotriangulations (combinatorics, rigidity, universality, expansiveness, extremeness), linear equilateral trees can't lock, unfolding 4D chains.

This class covers several interesting results about pointed pseudotriangulations:
  • Their original use for polygon ray shooting data structures.
  • Their (fixed) number of edges and faces (in contrast to triangulations).
  • Their minimal generic rigidity
  • Every planar minimally generically rigid graph can be drawn as a pointed pseudotriangulation (a kind of universality)
  • Why they work as an algorithm for the Carpenter's Rule Theorem: they have expansive motions after removing a convex-hull edge. (In particular, we'll review some lemmas from CDR.)
  • In fact, these expansive motions are the “extreme rays” (edges) of the cone of all expansive motions.

In addition, we cover the following questions and results:

  • Why do we use expansiveness? Both convenience and mathematical power.
  • Why can't the energy algorithm get stuck in a local minimum?
  • New result: linear equilateral trees can't lock
  • Old result: 4D (and higher-dimensional) open chains can't lock

Download Video: 360p, 720p

Handwritten notes, page 1/5[previous page][next page][PDF]

Handwritten notes, page 1/5[previous page][next page][PDF]

Slides, page 1/11[previous page][next page][PDF]

Slides, page 1/11[previous page][next page][PDF]

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