The Rhodonea Curves 
(which have polar equations r=a Sin[kθ])

f[θ_]=a*Sin[k*θ]*Cos[θ]
g[θ_]=a*Sin[k*θ]*Sin[θ]

a=1.5;k=.6;
ParametricPlot[{f[θ],g[θ]},{θ,0,12*Pi}]
Clear[a,k];

a=4;
tt=Table[k=1/2-1/(2n);ParametricPlot[{f[θ],g[θ]},{θ,0,4*n*Pi},PlotRange->{{-4.3,4.3},{-4.3,4.3}},PlotPoints->500,Background->RGBColor[1, 0.64, 0.28],ImageSize->350,AspectRatio->1]
,{n,1,30}];
Clear[a];

ListAnimate[tt]