Drawing a cyclogon

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,
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.

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