Simo K. Kivelä, 3 Nov 2007

Epicycloid and hypocycloid

A circle is rolling along a fixed circle without slipping. The locus of a point on the circumference of the rolling circle is called epicycloid, if the rolling circle is outside the fixed circle, and hypocycloid, if it is inside. The point forming the locus may also lie inside or outside of the rolling circle, in which case the locus is called epitrochoid or hypotrochoid. The parametric representation of the curve is

{(r+1) cos(t)-d cos((t/r)+t),(r+1) sin(t)-d sin((t/r)+t)},

where t is the curve parameter, d the distance of the moving point from the center of the rolling circle, r is the radius of the rolling circle and the radius of the fixed circle is 1.

In the following demonstration, the radius r of the moving circle and the distance d of the moving point can be changed. When the curve parameter is changed, the curve is plotted. If the radius of the moving circle is close to zero, the plot may be inaccurate. If so, adding the accuracy may help.

Manipulate[Graphics[{Circle[{0,0},1],{RGBColor[1,0,0],ParametricPlot[{(1+r) Cos[t]-d Cos[t/r+t],(1+r) Sin[t]-d Sin[t/r+t]},{t,0,b},MaxRecursion->n][[1,1,3,2]],PointSize[0.01],Point[{(1+r) Cos[b]-d Cos[b/r+b],(1+r) Sin[b]-d Sin[b/r+b]}]},ParametricPlot[{(1+r) Cos[b]+r Cos[t],(1+r) Sin[b]+r Sin[t]},{t,-Pi,Pi}][[1]],{PointSize[0.01],Point[{(1+r) Cos[b],(1+r) Sin[b]}]},Line[{{(1+r) Cos[b],(1+r) Sin[b]},{(1+r) Cos[b]-d Cos[b/r+b],(1+r) Sin[b]-d Sin[b/r+b]}}]},PlotRange->{{-4,4},{-4,4}},ImageSize->500,Axes->True],{{r,0.5,"radius of the moving circle"},-2,1,Appearance->"Labeled"},{{d,0.5,"distance of the moving point"},0,2,Appearance->"Labeled"},{{b,0.000001,"curve parameter"},0.000001,8 Pi,Appearance->"Labeled"},{{n,3,"accuracy"},3,10,1}]

Problems

1) For which values of r does the curve coincide with itself?

2) Are there values for d and r for which the curve is a segment of a line?

3) What happens if a) r=0, b) r=-1, c) r<-1?

4) On which condition for d and r does the curve meet the origin (i.e. the center of the fixed circle)?