newPursuit[k_, n_, scale_] := 
  Module[{}, 
   folf[l_List, i_] := 
    newptRadiusMid[##, i] & @@@ Partition[l, 3, 1, 2];
   midpt[pc_, pd_] := (Tr /@ Thread@{pc, pd})/2;
   newptRadiusMid[pa_, pc_, pd_, time_] := 
    Module[{pnew, xnew, ynew, pab, radius, pb}, pb = midpt[pc, pd];
     incr = 1; dist = EuclideanDistance[pa, pb];
     radius = If[dist > incr, incr, dist];
     pab = {pb[[1]] - pa[[1]], pb[[2]] - pa[[2]]};
     xnew = 
      If[pab[[1]] == 0, pa[[1]], 
       Sign[pab[[1]]] radius/
          Sqrt[1 + (pa[[2]] - pb[[2]])^2/(pa[[1]] - pb[[1]])^2] + 
        pa[[1]]];
     ynew = 
      If[pab[[1]] == 0, pa[[2]] + Sign[pab[[2]]] radius, 
       pa[[2]] + (xnew - pa[[1]]) #[[2]]/#[[1]] &@pab];
     pnew = {xnew, ynew}];
   abs[li_, p_] := 
    Module[{a, b, c, d, e, f, g}, 
     a = {Min@#, Max@#} &@li[[p]][[All, 1]];
     b = {Min@#, Max@#} &@li[[p]][[All, 2]];
     c = Abs[Differences@a];
     d = Abs[Differences@b];
     e = Max@{c, d}/2; f = 1.1; g = f*e;
     {{# - g, # + g} &@Mean@a, {# - g, # + g} &@Mean@b}];
   newlist = 
    Chop /@ FoldList[folf, scale*N@RandomSample[CirclePoints[k], k], 
      Range@n];
   plots = 
    Table[Graphics[{PointSize[
        Exp[1/Log[E + k] - 1]/(k/Log[k/2 + E] Log[Log[k + E^E]])], 
       Point /@ newlist[[m]]}, PlotRange -> abs[newlist, m], 
      Frame -> True], {m, 1, n, 1}];
   Manipulate[
    plots[[m]], {m, 1, n, 1, Appearance -> "Open", 
     AnimationRate -> k}]];
newPursuit[k_, n_] := newPursuit[k, n, k];
newPursuit[k_] := newPursuit[k, k Sqrt[k], k];

(*eg newPursuit[100]*)