Sinusoidal Spiral Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Initialization

sinuspiral=a{Cos[t]Cos[n t]^(1/n),Sin[t]Cos[n t]^(1/n)};

Equation

pedal=FullSimplify[PedalCurve[sinuspiral,{0,0},t]]

FullSimplify[Sqrt[#.#]&[pedal],a>0]

Plot

Off[ParametricPlot::"pptr"]

Show[GraphicsArray[{{
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->-2}],{0,0},{t,-1,1},{0,1,.15},DisplayFunction->Identity],
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->-1/2}],{0,0},{t,-2,2},{-Pi/2,Pi/2,.2},PlotRange->{{-.1,1.5},{-1,1}},DisplayFunction->Identity],
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->-1/3}],{0,0},{t,-4,4},{0,3.5,.3},DisplayFunction->Identity],
PedalCurvePlot[Limit[sinuspiral /.{a->1},n->0],{0,0},{t,-3,3},{0,2Pi,.2},DisplayFunction->Identity]},
{
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->1/2}],{0,0},{t,0,2Pi},{-Pi/2,Pi/2,.2},DisplayFunction->Identity],
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->2}],{0,0},{t,.01,2Pi},{.1,3,.1},DisplayFunction->Identity],
PedalCurvePlot[Evaluate[sinuspiral /.{a->1,n->3}],{0,0},{t,.01,2Pi},{.1,3,.1},DisplayFunction->Identity]
}
}]]