Woven Fabric Geometry

What is modelled

A Jacquard card has one row per pick and one hole position per warp end; a hole means the end is raised. The pattern of raised and lowered ends sets how long the threads float and how often they interlace. Where a warp thread crosses a weft it has to bend, so it is longer than the cloth it makes; the extra length is the crimp. Peirce described the bending with circular arcs and straight runs.

Equations

• Half-wave p = (l − Dθ) cosθ + D sinθ, h = (l − Dθ) sinθ + D (1 − cosθ) with D = d1 + d2 and h1 + h2 = D.
• Crimp c = (l − p)/p, scaled by the share of crossings that are interlacings.
• Cover K = K1 + K2 − K1K2 with K = d · threads per cm / 10.
• Mean float is the repeat divided by the number of changes per end.

Checks run in the page

• An explicit arc-and-line centreline against the Peirce length.
• A raster of the top view against the cover formula.
• The float length counted in the card against the closed form.
• The limits of straight and fully bent threads.

Honest simplifications

Peirce geometry is exact for the plain weave with round, incompressible yarns; here the crimp of a float weave is scaled by the interlacing density, which is only an estimate. Where the pitch is smaller than the sum of the yarn diameters the model limits the bend to 95 % of the pitch, as real yarns flatten. The balance between warp and weft is set by the loom tension, replaced here by one slider; and the balance between warp and weft is set by the loom tension, which is replaced by one slider.

• Press Five-shaft satin and see the mean float rise while the crimp falls.
• Press Open voile: the cover falls well below 100 %.
• Move the waviness share toward the weft and watch the warp straighten.