Yours is one of the best i've read! Thanks.
I've been looking at this for a couple of hours, and conclude that there is one major issue with finding a discrete answer.
You can extract the big circle circumference into a straight line all you like ( rolling your small circle & counting revs) but in this interpolation you could EQUALLY well extract a bigger circle, diameter D+d or radius D/2 +d ( if you must) straighten this out and roll your little circle underneath it, instead of ontop of it! Ha!
I can tell you, it's longer you get more revs associated with the locus of the point, P.
That's the clincher, because we're fiddling around with " frames of reference". This transformatiom, or Lorentz function is not sinusoidal as you may try to draw, because the circumference of the big circle everyone focuses upon, is smaller than the outer locus of any point on the little circle as it sweeps.
You can approximate the sinusoidal motion of each locus using the interpolation of the arc length for sin ( x) ( i.e. for distance pi, amplitude 1, the arc length is ~3.80221, remember you can't integrate it discretely). If you then multiple by nx pi radians you will always get the correct number of revs in space.
This also accounts for why, when positioned ( like Einstein) on the centre of the big circle, looking outwards at the little circle, you do not perceive that you are on a " point" nor " rotating-- you just see discrete rotations. For example, D=4 d=1 you see 5 complete rotations, as the point P ( arbitrary) on the small circle phase lags.
Hope this helps.

Yours is one of the best i've read! Thanks.

I've been looking at this for a couple of hours, and conclude that there is one major issue with finding a discrete answer.

You can extract the big circle circumference into a straight line all you like ( rolling your small circle & counting revs) but in this interpolation you could EQUALLY well extract a bigger circle, diameter D+d or radius D/2 +d ( if you must) straighten this out and roll your little circle underneath it, instead of ontop of it! Ha!

I can tell you, it's longer you get more revs associated with the locus of the point, P.

That's the clincher, because we're fiddling around with " frames of reference". This transformatiom, or Lorentz function is not sinusoidal as you may try to draw, because the circumference of the big circle everyone focuses upon, is smaller than the outer locus of any point on the little circle as it sweeps.

You can approximate the sinusoidal motion of each locus using the interpolation of the arc length for sin ( x) ( i.e. for distance pi, amplitude 1, the arc length is ~3.80221, remember you can't integrate it discretely). If you then multiple by nx pi radians you will always get the correct number of revs in space.

This also accounts for why, when positioned ( like Einstein) on the centre of the big circle, looking outwards at the little circle, you do not perceive that you are on a " point" nor " rotating-- you just see discrete rotations. For example, D=4 d=1 you see 5 complete rotations, as the point P ( arbitrary) on the small circle phase lags.

Hope this helps.