soln=Solve[{-2(1-L)+2 lambda Cos[theta]+mu Sin[theta]==0,
(1/2) theta-2 lambda L Sin[theta]+mu L Cos[theta]==0},{lambda, mu}]

Simplify[%]

s=Pi/64; m=24;

Do[
	f[k_]:=ParametricPlot[{Cos[k s] - L Cos[k s] + (k s Sin[k s])/(4 L),
 		-(k s Cos[k s])/(2 L) - 2 (-1 + L) Sin[k s]}, {L, .25, 1.75},
 		PlotRange -> {{.3, .6}, {-.05, .05}}],
{k,-m,m,1}];

Show[Join[Table[f[k], {k, -m, m, 1}]],AspectRatio->.6]