Comments by "Ficus-lovin\x27 Capybara N\x27 pals • 🌟 • 25 yrs ago" (@YourCapybaraAmigo_17yrsago) on "Veritasium"
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Even without watching this video through to the end, I intuitively understand why there would be up to one less rotation when Circle A is rotated along a flat surface equal to the circumference of circle B, and I even think I could explain it but I'm afraid it would come out too garbled to be easily understood. But visually, it's quite clear. It's because, in the flatline method, the two tangent points are always going to be closer together then the two sequentially tangent points would be on the two just-touching circles where both circles are curving away from each other at every point. There is literally less vertical and horizontal distance between each possible tangent point in the first scenario when circle A is being rotated along a straight line (of an unrolled, flattened Circle B).
Ergo, it's going to take measurably more rotation in the case of the two opposing circles, for Circle A to come back to its initial position on the edge of Circle B, versus traveling the same distance along a straight uncurved line equal to length of the circumference of Circle B.
I don't know if that makes any sense, I'm not a mathematician and I'm no special brainiac, but that's how I understand it.
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