Cardioid Negative Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equations

np=FullSimplify[NegativePedalCurve[a(1+Cos[t]){Cos[t],Sin[t]},{x,y},t]]

Function[t,a(1+Cos[t]){Cos[t],Sin[t]}]/@{0,Pi}

(0, 0)

FullSimplify[np/.{x->0,y->0}]

With[{a=1},ParametricPlot[{
{a (1+Cos[t]),a Sin[t]},
(1+Cos[t]){Cos[t],Sin[t]}
},{t,0,2Pi},AspectRatio->Automatic,PlotStyle->{Red,{}},Ticks->None]
]

(2a, 0)

FullSimplify[np/.{x->2a,y->0}/.t->2tt]/.tt->t/2

With[{a=1},ParametricPlot[{
{-(2/3) a ((Cos[(t/2)])^2),(2/3) a ((Sin[(t/2)])^2) Tan[(t/2)]},
(1+Cos[t]){Cos[t],Sin[t]}
},{t,-3.14,3.14},AspectRatio->Automatic,PlotStyle->{Red,{}},Ticks->None,
PlotRange->{{-1,2.1},{-1.8,1.8}}]
]