Here, we see the gradients at selected points as the "alpha" parameter for the noise pattern is animated. A the grid points, marked with orange dots as before, the gradient rotates in perfect sync with the wiggle, because near the grid points only the nearest wiggle has any impact on the noise value. However, between grid points, where there is overlap between the wiggles, the gradient behaves differently. It even rotates in the opposite direction in many places. This is good, because it hides the regularity and makes the motion more "random-looking".
Okay, we have a gradient that is cheap to compute, and it seems to behave nicely when "alpha" is animated. What can we do with that? As it turns out, we can do a lot.
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