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Is it possible to draw the path of a point on the boundary while the polygon is rolling, or writing a function of theta similar to the cycloid parametric equations:

$$ x = r (\theta - \sin(\theta)),\quad y = r (1 - \cos(\theta) $$

My main goal is to generalize the problem to write the parametric equations for the cyclogon. The following part of a program shows the multiple of four polygons after moving them above the x-axis and to the right of the y-axis. Please see below. The next step is to find a function to draw a path while a point on its vertex is rotating.

Manipulate[
  trM4 = 
    Graphics[
      {{EdgeForm[{Thick, Red}], FaceForm[LightGray], RegularPolygon[n]}, 
       {PointSize[0.025], Blue, Point[{Cos[π/n] - Sin[π/n], 0}]}, 
       If[hint, 
         Inset[
           Text @ Style[TraditionalForm[{Cos[π/( n)] - Sin[π/( n)], 0}], 16, Black], 
           {Cos[π/( n)] - Sin[π/n], -0.2}]], 
       {Translate[
          {EdgeForm[{Thick, Red}], FaceForm[{Yellow, Opacity[0.5]}], 
           RegularPolygon[n], {PointSize[0.03], Red, Point[{0, 0}]}}, 
          {Cos[π/n], Cos[π/n]}]}}, 
       Axes -> True, ImageSize -> 400];
  Show[trM4],
  Row[{
    Control[{{n, 4, "Number of sides"}, 4, 21, 1, Appearance -> "Labeled"}], 
    Spacer[45], 
    Control[{{hint, False, "hint"}, {False, True}}]}], 
  TrackedSymbols :> {n, hint}]

The code above creates polygons that start above the x-axis and to the left of the y-axis.

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  • 1
    $\begingroup$ Have you seen this and the references in the details? $\endgroup$ – C. E. Aug 30 at 19:58

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