Deltoid Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equation

deltoid={2/3 Cos[t]+1/3 Cos[2t],2/3 Sin[t]-1/3 Sin[2t]};

cat=Simplify[Catacaustic[{2Cos[t]+Cos[2t],2Sin[t]-Sin[2t]},{x,y},t]]

At origin

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

At any infinite point

FullSimplify[cat/.{x->r Cos[θ],y->r Sin[θ]}]

FullSimplify[Limit[%,r->Infinity]]

At x=∞

TrigReduce[FullSimplify[Limit[cat,x->∞]]]

At y=∞

FullSimplify[Limit[cat,y->∞]]

Plot

<<Graphics`Arrow`

movie=Table[ParametricPlot[{
{2Cos[t]+Cos[2t],2Sin[t]-Sin[2t]},
{3 Cos[t]-Cos[2 θ]+Cos[3 t+2 θ],3 Sin[t]+Sin[2 θ]-Sin[3 t+2 θ]}
},
{t,0,2Pi},
AspectRatio->Automatic,
Axes->False,
PlotStyle->{{},{Blue,Thickness[.015]}},
Epilog->{Red,Arrow[{0,0},-1{Cos[θ],Sin[θ]}]},
PlotRange->{{-1,1},{-1,1}}5
],{θ,0,2π-2Pi/50,2π/50}]

Show[GraphicsArray[Partition[Take[%161,{1,50,5}],5]]]

<<GIF`

Export[gifs<>"deltcaus.gif",movie]//Timing

Show[GraphicsArray[CatacausticPlot[{2Cos[t]+Cos[2t],2Sin[t]-Sin[2t]},#,{t,0,2π,2π/30},
DisplayFunction->Identity]&/@{{100000,0},{0,100000},{100000,100000}}
]]