Misc. Yarn cross-section and path length

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ldanielrosa
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TL;DR
How much does yarn 'squish'?
I'm trying to find the "ideal" cross-section for woven yarns. As seen in the first image, a circular monofilament (assuming inviolable cross-section but infinite flexibility) will have a limit of 2d thickness, and repeat period of 2d√3.
IMG_20251127_163557.webp

This allows me to calculate how long the yarn path length is, whether I want to know the weight or how much yarn to supply for a given amount of product.

However when the yarn is untwisted multifilament, the cross section will take a different shape. I'm having trouble determining exactly what is the optimum cross section.

IMG_20251127_163612.webp


IMG_20251127_163627.webp

IMG_20251127_163642.webp



I do not know whether the shape will be governed by circular arcs, parabolic arches, sinusoidal form, or something else entirely. I am assuming that the football/ellipsoid will be symmetrical across a line drawn between corners because this will equalize strain, but what I can intelligently guess ends there.

I want to know not only because I want to be able to calculate the path length for the reasons mentioned above (materials required and weight), but also because I want to know the thickness and pore size. My application is on a tubular braider (16 carrier), and soon on a 24 carrier braider, and 17 and 33 carrier flat braiders as well.

Can someone please direct me to an article I can read that will explain this?
 
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