pursuit[k_, n_, 
   scale_] := (midpt[pc_, pd_] := (Tr /@ Thread@{pc, pd})/2;
   newpt[pa_, pc_, pd_] := Module[{pb, pnew, xnew, ynew, pab},
     pb = midpt[pc, pd];
     pab = {pb[[1]] - pa[[1]], pb[[2]] - pa[[2]]};
     xnew = 
      If[pab[[1]] == 0, pa[[1]], 
       Sign[pab[[1]]] 1/
          Sqrt[1 + (pa[[2]] - pb[[2]])^2/(pa[[1]] - pb[[1]])^2] + 
        pa[[1]]];
     ynew = 
      If[pab[[1]] == 0, pa[[2]] + Sign[pab[[2]]] 1, 
       pa[[2]] + (xnew - pa[[1]]) #[[2]]/#[[1]] &@pab];
     pnew = 
      If[EuclideanDistance[pa, pb] < 
        EuclideanDistance[pa, {xnew, ynew}], pb, {xnew, ynew}]];
   t3[li_] := 
    Take[#, 3] &@RotateLeft[li, #] & /@ Range[-1, Length@li - 2];
   abs[p_] := Module[{a, b, c, d, e, f, g},
     a = {Min@#, Max@#} &@li[[p, 1]][[All, 1]];
     b = {Min@#, Max@#} &@li[[p, 1]][[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}];
   list = RandomSample[(scale)*CirclePoints[k], k];
   li = Thread[{Quiet[
       FoldList[
        Chop /@ N /@ newpt[#[[1]], #[[2]], #[[3]]] & /@ t3@# &, 
        N@list, Range@n]]}];
   ptable = 
    Table[Graphics[{PointSize[
        Exp[1/Log[E + k] - 1]/(k/Log[k/2 + E] Log[Log[k + E^E]])], 
       Point /@ li[[m]][[1]]}, Frame -> True, PlotRange -> abs@m], {m,
       1, n, 1}];
   Manipulate[
    ptable[[t]], {t, 1, n, 1, Appearance -> "Open", 
     AnimationRate -> k}]);
pursuit[k_, n_] := pursuit[k, n, 1]; 
pursuit[k_] := pursuit[k, Floor[Sqrt[k] k], 1];
(*
example implementation ~;
pursuit[100];
pursuit[#,Floor[Sqrt[#]#],#]&[100];
pursuit[#,Floor[2Sqrt[#]#],# Log[#]]&[50];
pursuit[#,Floor[Sqrt[#]#],1]&[200];
ListAnimate[ptable[[1;;-1;;20]],AnimationRate\[Rule]100];
(Differences/@abs@#)[[1,1]]&/@Range@Length@ptable//ListLinePlot
*)