Takahashi, Kohske. "Curvature Blindness Illusion." i-Perception 8.6 (2017): 2041669517742178. (Journal link)
Q. I wonder if there is some mathematical hypothesis (as opposed to a strictly physiological hypothesis) that can help explain this remarkable (newly discovered) illusion?
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My percept matches what you would get if you replaced each segment with its best-fitting circular arc.
For the zigzag case the best-fitting circular arc has zero curvature, i.e. it is a straight line segment.
If this hypothesis is correct, in the other case the curve should appear more "rounded" than a usual sine wave, and to be honest it kind of does.
Looks simple enough to me. The darker bands are interpreted as shadows when they end right in the middle of the curve top or bottom, and that makes you reinterpret the curve as a corner (because the "shadow" ends so abruptly. And conversely, when the darker band continues through the middle, your brain will not interpret it as a corner because it knows that shadows are not in the habit of turning corners like that. At least that's my first impression.