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Epicycloid and hypocycloid

Simo K. Kivelš
Organization: Helsinki University of Technology
Department: Institute of Mathematics
Education level


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.

*Education > College

EpicycloidAndHypocycloid.nbp (802.3 KB) - Mathematica Player Notebook