Triangle strip knitting
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Triangle Strip Knitting James Mallos ISAMA 2010 Chicago, Illinois
description
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.
Transcript of Triangle strip knitting
Triangle Strip Knitting
James MallosISAMA 2010
Chicago, Illinois
Can we knit (or crochet) any surface?
• One yarn.
• One piece (work in progress must remain a single, well-connected piece.)
• One technique (sewing up seams is not allowed.)
Rules
We already know weaving can do this.
Akleman, E., Chen, J., Xing, Q., and, Gross, J. ’2009.
Every polygonal surface mesh describes a plain-weaving.
The Plain-Weaving Theorem
A consequence of PWT: every tessellation of the plane describes a plain-woven fabric.
The PWT can be demonstrated with a special set of Truchet tiles.
A virtual Truchet tiling can be done using the computer graphics technique of texture mapping.
Does a small set of polygons suffice (just triangles, say)?
No...not if we want the boundaries of the basket to have selvaged edges. Boundaries are, in effect, large n-gons that need to be tiled like all the other polygons.
A 3D model decorated with virtual Truchet tiles.
Model courtesy INRIA via the Aim@Shape Shape Repository.
Offering, James Mallos, 2008
A woven sculpture derived from a surface mesh.
Model courtesy INRIA via the Aim@Shape Shape Repository.
Olivier’s Fingertip, James Mallos, 2008
A woven sculpture derived from a surface mesh.
Model courtesy INRIA via the Aim@Shape Shape Repository.
Big Little, James Mallos, 2010
A woven sculpture derived from a surface mesh.
Model courtesy INRIA via the Aim@Shape Shape Repository.
Can knitting and crochet also make any surface?
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 manipulations of the unknot.
Since they are manipulations of the unknot, K & C can be done with the ends of the yarn tied together.
In practice, this adds no difficulty.
Because they are manipulations of the unknot, 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 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.)
how would you mow the grass on this planet?
Finding a linear order of working that covers the surface:
Three Ways to Mow Grass
Boustrophedonic SpiralSerpentine Loop
(Traveling Salesman)
Which way works on a general surface?
Any compact surface can be mapped onto the interior of a plane polygon—the topological complexities are confined to the way the polygon edges identify in pairs.
A method of cutting grass in the interior of a plane polygon (without crossing the perimeter) will map onto any surface.
They all do!
Of the three mowing schemes, only the Serpentine Loop is versatile.
Boustrophedonic SpiralSerpentine Loop
(Traveling Salesman)
Got an obstacle? Take cities in that region off the salesman’s list. Need more refinement somewhere? Add more cities there.
A triangle strip corresponds to a Hamiltonian Cycle (TSP solution) on the dual graph of the triangulation.
Gopi and Eppstein 2004
Example of a Hamiltonian cycle on the dodecahedron (dual to the icosahedron.)
• 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.)
Hamiltonian Facts of Life
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.
Good news: If we don’t find a Hamiltonian cycle, we can make one!
• The Single Strip Algorithm can be made to respect constraints such as preferred directions.
Gopi and Eppstein 2004
The Single-Strip Algoritm gives us a strip (or loop) of triangles, how do we knit and assemble a strip of triangles?
• 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.
There are four kinds of vertex in the hamiltonian cycle that can be labelled in this way:
• 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
undp
Four “emoticons” can naturally represent the four labels:open left
open rightclose left
close right
• Tetrahedron: undp and nupd
• Octahedron: unnduppd and nuupnddp
• Icosahedron: nnununuuupppdpdndddp
Some Undip Codewords for Deltahedra
Codewords suffice for genus 0 surfaces only. Higher genus surfaces need more information.
Note:
TRIANGLE CONTEXT CHART
CAST OFF
d p
u n
CLOSE LEFT CLOSE RIGHT
OPEN LEFT OPEN RIGHT
CAST ON
• We want correctly imbedded surfaces.
• Correct Gaussian curvature (intrinsic curvature) is necessary but not sufficient.
• Correct topology is necessary but not sufficient.
Caveat:
