\documentclass{standalone}
\usepackage{luamplib}
\begin{document}
\mplibtextextlabel{enable}
\begin{mplibcode}
beginfig(1);
numeric R, r, pi;
R = 1.8cm; % radius of the outer part 
r = 1.3cm; % radius of the inner part
pi = 3.14159265; % constant
% define the cycloid
path c; 
c = (0,-R) rotated 100 shifted (r*-100/180*pi,r) 
    for t=-99 upto 460: 
      -- (0,-R) rotated -t shifted (r*t/180*pi,r) 
    endfor;
% axes
drawoptions(withcolor .5 white);
path xx, yy;
yy = (down -- 5 up) scaled 1/2 r;
xx = (xpart point 0 of c, 0) -- (xpart point infinity of c,0);
draw fullcircle scaled 1/4r; drawarrow xx; drawarrow yy;
drawoptions();
label.rt (btex $x$ etex, point 1 of xx);
label.top(btex $y$ etex, point 1 of yy);

% draw the cycloid on top of the axes
draw c withcolor .67 red;

% define a couple of related points: 
% z1 center of the blue wheel
% z2 intersection of rim and cycloid
t = 124; % if you change t then the wheel will "roll" along...
z1 = (r*t/180*pi,r);
z2 = (0,-R) rotated -t shifted z1;

% draw the auxiliary lines
draw (0,y2) -- z2 -- (x2,0) dashed withdots scaled .6;
draw z2 -- z1 -- (x1,0);
% draw the rolling circle and mark centre and intersection 
draw fullcircle scaled 2r shifted z1 withcolor 2/3 blue;
draw fullcircle scaled 2R shifted z1 withcolor 1/2[2/3 blue, white];
fill fullcircle scaled dotlabeldiam shifted z1; 
fill fullcircle scaled dotlabeldiam shifted z2;
% some arc arrows and labels
path a[];
z3 = (x1,5/12y1); 
a1 = z3 {left} .. {left rotatedabout(z1,-t)} z3 rotatedabout(z1,-t);
drawarrow subpath (.05,.95) of a1;
label.llft(btex $\theta$ etex, point .5 of a1);
a2 = subpath (0,1) of reverse quartercircle scaled 2.2r shifted z1;
drawarrow a2 rotatedabout(z1,-100);
% finally all the other labels
label.rt (btex $r$ etex, (x1,.5y1));
label.urt(btex $R$ etex, .6[z1,z2]);
label.lft(btex $y$ etex, (0,y2));
% give all the x-axis labels a common baseline with mathstrut
label.bot(btex $\mathstrut x$ etex, (x2,0));
label.bot(btex $\mathstrut r\theta$ etex, (x1,0));
label.bot(btex $\mathstrut 2\pi r$ etex, (r*2pi,0));
draw (down--up) scaled 2 shifted (r*2pi,0) withcolor .5 white;
% notice how nicely the coordinates work...
dotlabel.top(btex $(\pi r,R+r)$ etex, (pi*r,R+r));
% and a little alignment to finish
label(btex $\vcenter{\halign{&$#$\hfil\cr 
x=r\theta-R\sin\theta\cr
y=r-R\cos\theta\cr}}$ etex,(4.75r,r));
endfig;
\end{mplibcode}
\end{document}
