Sorry it's taking me a bit of time to catch up here.

Last week I promised to show how the effective transparency of opaque vertical tubes was calculated.

Here's the first of two diagrams.

The circles marked A and B are two adjacent fibers or tubes, viewed in cross section. Since the cloth repeats this over and over, we need only consider the effective opacity between the center points A and B.

Imagine the camera is at the bottom, looking up.

The fibers are separated by some distance, which we'll call S. In this case, I made S = 1, but it really doesn't matter what it is because everything involved is going to be a ratio.

Now recall that I defined the "density" (D) of the fibers as a fraction of the spacing (S).

In this case, I'm using a Density of .4 or 40% of the spacing. Which means the radius of each fiber is half that, or .2. But between A and B we have an obstruction from both fibers, resulting in a total obstruction of D. 

The red bar, marked T, is the area of transmittance - the gap we see between the fibers.

So the ratio of obstruction (which is the defintiion of opacity here) is precisely D, The shaded blue bars show us the area of obstruction in the space between A and B.

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Renderosity forum reply notifications are wonky. If I read a follow-up in a thread, but I don't myself reply, then notifications no longer happen AT ALL on that thread. So if I seem to be ignoring a question, that's why. (Updated September 23, 2019)