Epicycloid Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/EpicycloidPedalCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/EpicycloidPedalCurve.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

Epicycloid[{a,b},t]

PedalCurve[Epicycloid[{a,b},t],{x0,y0},t]//FullSimplify

PedalCurve[Epicycloid[{a,b},t],{0,0},t]//FullSimplify

ped=PedalCurve[Epicycloid[{n,1},t],{0,0},t]//FullSimplify

FullSimplify[Sqrt[ped.ped],0<t<2π&&n>1]

FullSimplify[ArcTan[Divide@@Reverse[ped]]]

Solve[θ==-ArcTan[Cot[(1/2) (2+n) t]],t]

Sin[((n t)/2)]/.t->-((2 ArcCot[Tan[θ]])/(2+n))

Plot

<<Utilities`Typesetting`

Off[Power::"infy",∞::"indet",Graphics::"gptn"];

Show[GraphicsArray[Table[
PedalCurvePlot[Epicycloid[{n,1},t],{0,0},{t,0,2Pi},{0,Pi/2,.1},
DisplayFunction->Identity,PlotLabel->TraditionalForm[HoldForm[n]==n]],{n,4}]
]]

Show[GraphicsArray[
Table[PolarPlot[Sin[(n/(n+2))(t+Pi/2)],{t,0,(n+2)π },PlotStyle->Red,PlotLabel->TraditionalForm[HoldForm[n]==n],DisplayFunction->Identity],{n,4}]
]]