numeric r, pi; r = 1.25cm; % radius of the wheel 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; % and the 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 = 84; % 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 draw fullcircle scaled 2r shifted z1 withcolor .67 blue; % mark the centre and intersection with cycloid 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); drawarrow a2 rotatedabout(z1,80); % finally all the other labels label.top(btex $r$ etex, .5[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)); % notice how nicely the coordinates work... dotlabel.top(btex $(\pi r,2r)$ etex, (pi*r,2r)); % and a little alignment to finish label(btex \vcenter{\halign{&#$\hfil\cr x=r(\theta-\sin\theta)\cr y=r(1-\cos\theta)\cr}}$ etex,(4.2r,r));