Simo K. Kivelä, 3 Nov 2007

Cycloid

A circle is rolling along a staight line without slipping. The locus of a point on the circumference of the rolling circle is called cycloid. If the point is inside or outside of the rolling circle, the locus is called trochoid. The parametric representation of the curve is

{t-d sin(t),1-d cos(t)},

where t is the curve parameter, d the distance of the moving point from the center of the rolling circle and the radius of the rolling circle is 1. If d=1, the curve is cycloid, otherwise trochoid.

In the following demonstration, the distance d can be changed. When the curve parameter is changed, the curve — cycloid or trochoid — is plotted.

Manipulate[Graphics[{ParametricPlot[{t-d Sin[t],1-d Cos[t]},{t,0,b}][[1]],PointSize[0.01],Point[{b-d Sin[b],1-d Cos[b]}],Circle[{b,1},1],Line[{{b,1},{b-d Sin[b],1-d Cos[b]}}]},PlotRange->{{-2,27},{-3,5}},ImageSize->700,Axes->True],{{d,1,"distance of the moving point"},0,3,Appearance->"Labeled"},{{b,0.0001,"curve parameter"},0.0001,8 Pi,Appearance->"Labeled"}]

Problems

1) For which values of d does the curve define y as a function of x?

2) Compute the coordinates for the lowest points of the curve.

3) For some values of d, the curve intersects itself. Find the coordinates of the intersection points (as a function of d, if possible; if not, give the method, e.g. a piece of computer code).

4) Find the angle between the tangents of the curve at the points where the curve intersects itself.