\documentclass[border=5mm]{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
   numeric alpha; alpha = 34;
   transform t;
   origin transformed t = origin;
   right transformed t = dir -alpha;
   up transformed t = dir alpha;

   path xx, yy;
   xx = (origin--right) scaled 144;
   yy = (origin--up) scaled 144;

   numeric a, s; s = 1/64; a = 3/2;
   path hh;
   for x = 8 * a * s step s until 8/a:
       hh := if known hh: hh ... fi (a * x, a / x);
   endfor;
   hh := hh scaled 16;
   path C; C = fullcircle scaled 2a scaled sqrt(2) scaled 16;
   pair F; F = (xpart point 0 of C, ypart point 2 of C);

   interim ahangle := 30;
   picture P[];
   P1 = image(
       draw point 0 of C -- F -- point 2 of C withcolor 3/4[blue, white];
       draw subpath (-1/2, 5/2) of C withcolor 3/4[blue, white];
       drawarrow xx withcolor 1/2;
       drawarrow yy withcolor 1/2;
       draw hh withcolor 2/3 red;  dotlabel.urt("$S$", F);
   );
   P2 = image(
       hh := hh transformed t;
       C := fullcircle scaled 2 abs(point directiontime down of hh of hh);
       F := F rotated -45;
       (p, whatever) = C intersectiontimes (fullcircle scaled abs(F) shifted 1/2 F);
       draw point p of C -- F -- point -p of C withcolor 3/4[blue, white];
       draw subpath (-alpha/30, alpha/30) of C withcolor 3/4[blue, white];
       drawarrow xx transformed t withcolor 1/2;
       drawarrow yy transformed t withcolor 1/2;
       draw hh withcolor 2/3 red;
       dotlabel.urt("$S$", F);
   );

   label.lft(P1, 10 left);
   label.rt(P2, 10 right);



   %path C;
   %C = fullcircle scaled 4 cosd(alpha) scaled 32a ;
   %draw C withcolor 1/2;

   %pair S;
   %S = whatever[precontrol +alpha/45 of C, postcontrol +alpha/45 of C]
   %  = whatever[precontrol -alpha/45 of C, postcontrol -alpha/45 of C];
   %draw (64a * right) withpen pencircle scaled 3 withcolor 3/4[blue, white];
   %draw S withpen pencircle scaled 3/2 withcolor red;

   %draw hh withcolor 2/3 red;
   %%draw hh rotated 180 withcolor 2/3 red;

   %% path t;
   %% t = point 4 of hh -- point 80 of hh -- point 400 of hh -- cycle;

   %% pair a, b, c;
   %% a = whatever[point 0 of t, point 1 of t]; point 2 of t - a = whatever * (point 0 of t-point 1 of t) rotated 90;
   %% b = whatever[point 2 of t, point 1 of t]; point 0 of t - b = whatever * (point 2 of t-point 1 of t) rotated 90;

   %% draw t;  c = whatever[point 2 of t, a] = whatever[point 0 of t, b];

   %% draw c withpen pencircle scaled 3;



endfig;
\end{mplibcode}
\end{document}