Abstract:
We present a bicontinuous, minimal surface (the helicoid) as a scaffold on
which to define the topology and geometry of yarns in a weft-knitted fabric.
Modeling with helicoids offers a geometric approach to simulating a physical
manufacturing process, which should generate geometric models suitable for
downstream mechanical and validity analyses. The centerline of a yarn in a
knitted fabric is specified as a geodesic path, with constrained boundary
conditions, running along a helicoid at a fixed distance. The shape of the
yarn’s centerline is produced via an optimization process over a polyline.
The distances between the vertices of the polyline are shortened and a
repulsive potential keeps the vertices at a specified distance from the
helicoid. These actions and constraints are formulated into a single
"energy" function, which is then minimized. The yarn geometry is generated
as a tube around the centerline. The optimized configuration, defined for a
half loop, is duplicated, reflected, and shifted to produce the centerlines
for the multiple stitches that make up a fabric. In addition, the parameters
of the helicoid may be used to control the size and shape of the fabric's
stitches. We show that helicoid scaffolds may be used to define both knit
and purl stitches, which are then combined to produce models of all-knit,
rib, and garter fabrics.
