Ygtjki

Further to this question I found on MSE, I tried to replicate

from here

this is as far as I got:

fun[a_, b_, c_, x_, y_] := 
 Point[#[[1]] + x, #[[2]] + y &[
 Part[CirclePoints[360] c, 
 If[a + b == 360, 360, Mod[a + b, 360]]]]];
tab = With[a = #, 
 Flatten[Table[
 Table[fun[a, 90 + 15 n, 1 - .15 m, -1 + .5 n, -.35 m], m, 0, 
 10], n, 0, 24], 1]] & /@ Range[1, 360, 15];

Module[t, x, y, fun, xf, yf, a, x = -.5; y = 1;
 fun[a_, b_, c_, x_, y_] := 
 Point[#[[1]] + x, #[[2]] + y &[
 Part[CirclePoints[360] c, 
 If[a + b == 360, 360, Mod[a + b, 360]]]]];
 xf[t_, a_, b_] := a t - b Sin[t]; yf[t_, a_, b_] := a - b Cos[t];
 Animate[
 Show[
 Graphics[
 PointSize[.01], tab[[a]],
 PlotRange -> -1 - x, 10 + x, -1 - y, 1
 ],
 ParametricPlot[
 (Pi/2) xf[t + 2 Pi a/24, 1.25, .6] - 4 Pi a/24 - Pi^2 + .05, 
 2.05 - 1.65 yf[t + 2 Pi a/24, 1.25, .6],
 t, -4 Pi, 4 Pi, Axes -> False
 ]
 ],
 a, 1, 24, 1, ControlPlacement -> Top, AnimationRate -> 5, 
 AnimationDirection -> Backward
 ]
 ]

which is not very efficient (I'm sure Part could be applied more efficiently), and despite various tweeks, I couldn't quite manage to get the cycloid to line up with the points.

What is a better way to approach this?

DynamicModule[t = 0, d = 5, a = .08, base, distortion, pts, r, f, n = 10,

 r[y_] := .08 y^4;
 f[x_] := -2 Pi Dynamic[t] + d x; 
 (*f does not evaluate to a number but FE will take care of that later*)

 base = Array[List, n 3, 1, 0, Pi, 0, 1 ];

 distortion = Array[ 
 Function[x, y, r[y] Cos @ f @ x, Sin @ f @ x], n 3, 1, 0, Pi, 0, 1 
 ];

 pts = base + distortion;

 Row[
 Animator[Dynamic @ t, AnimationRate -> .8, AppearanceElements -> ],
 Graphics[ 
 LightBlue,
 Polygon @ Join[ pts[[;; , -1]], Scaled[1, 0], Scaled[0, 0]],

 Darker @ Blue, AbsolutePointSize @ 5, Point @ Catenate @ pts,

 AbsolutePointSize @ 7, Orange, Thick,
 Point @ pts[[15, -1]], Circle[base[[15, -1]], r @ base[[15, -1, 2]]],
 Point @ pts[[15, 7]], Circle[base[[15, 7]], r @ base[[15, 7, 2]]] 
 ,
 PlotRange -> 0 + .1, Pi - .1, 0, 1.2, 
 PlotRangePadding -> 0,
 PlotRangeClipping -> True, ImageSize -> 800]
 ]
 ]