OK. I have the first set of images ready to demonstrate the first part. There are actually going to be 4 parts. They get progressively harder to understand.

Ready?

Let's consider cloth fibers that are 100% opaque. Let's also consider only the vertical fibers arranged in a cloth that is straight vertically, but curved horizontally. We can model these using a bunch of tubes arranged in a semi-circle, like this.

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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)