Triangle Strip Knitting        James Mallos        ISAMA 2010       Chicago, Illinois
Can we knit (or crochet) any surface?
Rules• One yarn.• One piece (work in progress must remain a  single, well-connected piece.)• One technique (sewing up seam...
We already know weaving can do this.
The Plain-Weaving TheoremEvery polygonal surface meshdescribes a plain-weaving. Akleman, E., Chen, J., Xing, Q., and, Gros...
A consequence of PWT: every tessellation ofthe plane describes a plain-woven fabric.
The PWT can be demonstrated with a specialset of Truchet tiles.
A virtual Truchet tiling can be done using thecomputer graphics technique of texture mapping.
Does a small set of polygons suffice (justtriangles, say)?
No...not if we want the boundaries of thebasket to have selvaged edges. Boundariesare, in effect, large n-gons that need t...
Model courtesy INRIA                                        via the Aim@Shape                                        Shape...
Model courtesy                                          INRIA via the                                          Aim@Shape S...
Model courtesy                                               INRIA via the                                               A...
Model courtesy                                                  INRIA via the                                             ...
Can knitting and crochet also make anysurface?What’s the difference, K & C vs. W?
• Weaving is a multicomponent link, or  sometimes a single-component link (a knot).• Knitting and crochet are manipulation...
Since they are manipulations of the unknot, K& C can be done with the ends of the yarntied together.In practice, this adds...
Because they are manipulations of theunknot, C & K unravel. W does not.
• W has rotational symmetry around its  openings (a fact which makes Truchet tiles  easy use)• K and C do not have rotatio...
Finding a linear order of working that covers the                     surface:        how would you mow the grass         ...
Three Ways to Mow Grass                                    Serpentine LoopBoustrophedonic       Spiral                    ...
Which way works on a general surface?
They all do!Any compact surface can be mapped onto theinterior of a plane polygon—the topologicalcomplexities are confined ...
Of the three mowing schemes, only theSerpentine Loop is versatile.                                       Serpentine LoopBo...
Gopi and Eppstein 2004A triangle strip corresponds to a Hamiltonian Cycle(TSP solution) on the dual graph of the triangula...
Example of a Hamiltonian cycle on thedodecahedron (dual to the icosahedron.)
Hamiltonian Facts of Life• Nearly all triangulations without boundaries  have Hamiltonian duals.• If more than 15% of the ...
Good news: If we don’t find a Hamiltonian cycle,              we can make one!      The Single Strip Algorithm (Gopi and   ...
• The Single Strip Algorithm can be made to  respect constraints such as preferred  directions.                           ...
The Single-Strip Algoritm gives us a strip(or loop) of triangles, how do we knit andassemble a strip of triangles?
There are four kinds of vertex in the hamiltonian     cycle that can be labelled in this way:  • Arbitrarily choose a mid-...
• Label each vertex according to whether the  non-Hamiltonian edge extends to the left or  the right, and...• whether the ...
Four “emoticons” can naturally represent the four labels:                       open left                      open right ...
Some Undip Codewords for Deltahedra• Tetrahedron: undp and nupd• Octahedron: unnduppd and nuupnddp• Icosahedron: nnununuuu...
Note:Codewords suffice for genus 0 surfaces only.Higher genus surfaces need more information.
TRIANGLE CONTEXT CHART               CAST OFF    d                       p  CLOSE LEFT              CLOSE RIGHT    u      ...
Caveat:• We want correctly imbedded surfaces.• Correct Gaussian curvature (intrinsic  curvature) is necessary but not suffi...
Upcoming SlideShare
Loading in …5
×

Triangle strip knitting

1,183 views

Published on

A theoretical consideration of a method that can be used to knit or crochet an arbitrary surface. The method is based on the triangle-strip method in computer graphics. A talk presented at ISAMA 2010 in Chicago, IL.

Published in: Design, Technology
0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
1,183
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
30
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Triangle strip knitting

  1. 1. Triangle Strip Knitting James Mallos ISAMA 2010 Chicago, Illinois
  2. 2. Can we knit (or crochet) any surface?
  3. 3. Rules• One yarn.• One piece (work in progress must remain a single, well-connected piece.)• One technique (sewing up seams is not allowed.)
  4. 4. We already know weaving can do this.
  5. 5. The Plain-Weaving TheoremEvery polygonal surface meshdescribes a plain-weaving. Akleman, E., Chen, J., Xing, Q., and, Gross, J. ’2009.
  6. 6. A consequence of PWT: every tessellation ofthe plane describes a plain-woven fabric.
  7. 7. The PWT can be demonstrated with a specialset of Truchet tiles.
  8. 8. A virtual Truchet tiling can be done using thecomputer graphics technique of texture mapping.
  9. 9. Does a small set of polygons suffice (justtriangles, say)?
  10. 10. No...not if we want the boundaries of thebasket to have selvaged edges. Boundariesare, in effect, large n-gons that need to be tiledlike all the other polygons.
  11. 11. Model courtesy INRIA via the Aim@Shape Shape Repository.A 3D model decorated with virtual Truchet tiles.
  12. 12. Model courtesy INRIA via the Aim@Shape Shape Repository. Offering, James Mallos, 2008A woven sculpture derived from a surface mesh.
  13. 13. Model courtesy INRIA via the Aim@Shape Shape Repository. Olivier’s Fingertip, James Mallos, 2008A woven sculpture derived from a surface mesh.
  14. 14. Model courtesy INRIA via the Aim@Shape Shape Repository. Big Little, James Mallos, 2010A woven sculpture derived from a surface mesh.
  15. 15. Can knitting and crochet also make anysurface?What’s the difference, K & C vs. W?
  16. 16. • Weaving is a multicomponent link, or sometimes a single-component link (a knot).• Knitting and crochet are manipulations of the unknot.
  17. 17. Since they are manipulations of the unknot, K& C can be done with the ends of the yarntied together.In practice, this adds no difficulty.
  18. 18. Because they are manipulations of theunknot, C & K unravel. W does not.
  19. 19. • W has rotational symmetry around its openings (a fact which makes Truchet tiles easy use)• K and C do not have rotational symmetry: every K-tile or C-tile must be properly oriented inside its n-gon.• W does not reveal its order of working, but K and C do (K-tiles and C-tiles must align in a linear pattern that covers the surface.)
  20. 20. Finding a linear order of working that covers the surface: how would you mow the grass on this planet?
  21. 21. Three Ways to Mow Grass Serpentine LoopBoustrophedonic Spiral (Traveling Salesman)
  22. 22. Which way works on a general surface?
  23. 23. They all do!Any compact surface can be mapped onto theinterior of a plane polygon—the topologicalcomplexities are confined to the way thepolygon edges identify in pairs.A method of cutting grass in the interior of aplane polygon (without crossing the perimeter)will map onto any surface.
  24. 24. Of the three mowing schemes, only theSerpentine Loop is versatile. Serpentine LoopBoustrophedonic Spiral (Traveling Salesman)Got an obstacle? Take cities in that region off thesalesman’s list. Need more refinement somewhere?Add more cities there.
  25. 25. Gopi and Eppstein 2004A triangle strip corresponds to a Hamiltonian Cycle(TSP solution) on the dual graph of the triangulation.
  26. 26. Example of a Hamiltonian cycle on thedodecahedron (dual to the icosahedron.)
  27. 27. Hamiltonian Facts of Life• Nearly all triangulations without boundaries have Hamiltonian duals.• If more than 15% of the triangles are on boundaries (and therefore 2-valent), the dual is unlikely to be Hamiltonian.• Searching for a Hamilton path or circuit in the dual cubic graph becomes intractable for large triangulations. (NP complete.)
  28. 28. Good news: If we don’t find a Hamiltonian cycle, we can make one! The Single Strip Algorithm (Gopi and Eppstein, 2004) • Don’t try to find a Hamilton circuit, make one by gently editing the triangulation at a few points.
  29. 29. • The Single Strip Algorithm can be made to respect constraints such as preferred directions. Gopi and Eppstein 2004
  30. 30. The Single-Strip Algoritm gives us a strip(or loop) of triangles, how do we knit andassemble a strip of triangles?
  31. 31. There are four kinds of vertex in the hamiltonian cycle that can be labelled in this way: • Arbitrarily choose a mid-edge in the Hamiltonian Cycle as a starting point. • Arbitrarily choose a side of the surface at the starting point. • Arbitrarily choose a direction of travel.
  32. 32. • Label each vertex according to whether the non-Hamiltonian edge extends to the left or the right, and...• whether the adjacent vertex on the non- Hamiltonian edge has already been labelled (close) or not (open.)• Finish when the starting point is encountered
  33. 33. Four “emoticons” can naturally represent the four labels: open left open right close left close right undp
  34. 34. Some Undip Codewords for Deltahedra• Tetrahedron: undp and nupd• Octahedron: unnduppd and nuupnddp• Icosahedron: nnununuuupppdpdndddp
  35. 35. Note:Codewords suffice for genus 0 surfaces only.Higher genus surfaces need more information.
  36. 36. TRIANGLE CONTEXT CHART CAST OFF d p CLOSE LEFT CLOSE RIGHT u n CAST ON OPEN LEFT OPEN RIGHT
  37. 37. Caveat:• We want correctly imbedded surfaces.• Correct Gaussian curvature (intrinsic curvature) is necessary but not sufficient.• Correct topology is necessary but not sufficient.

×