Astroid Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Catacaustics for Various Radiant Points

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

General

Timing[cat=FullSimplify[Catacaustic[{Cos[t]^3,Sin[t]^3},{x,y},t]]]

y=0

FullSimplify[cat/.y->0]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},{1,0},{t,0,2π,2π/100}]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},{1/2,0},{t,0,2π,2π/100}]

(x,y)=(1,0)

FullSimplify[cat/.{x->1,y->0}]

ParametricPlot[{(Cos[t] (7+6 Cos[t]+4 Cos[2 t]+4 Cos[3 t]+3 Cos[4 t]))/(11+10 Cos[t]+3 Cos[2 t]),((4 (5+4 Cos[t]+3 Cos[2 t]) ((Sin[t])^3))/(11+10 Cos[t]+3 Cos[2 t]))},
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic]

y=∞

FullSimplify[Limit[cat,y->∞]]

ParametricPlot[{-(1/2) ((Cos[t])^3) (-5+3 Cos[2 t]),(1/16) (12 Sin[t]-Sin[3 t]+3 Sin[5 t])},
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic]

x=y

FullSimplify[Limit[cat/.{y->r,x->r},r->∞]]

ParametricPlot[{(1/16) (15 Cos[t]+Cos[3 t]+3 Sin[t]-3 Sin[5 t]),(1/16) (3 Cos[t]-3 Cos[5 t]+15 Sin[t]-Sin[3 t])},
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},.3{1,1},{t,0,2π,2π/100}]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},3{1,1},{t,0,2π,2π/100}]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},10{1,1},{t,0,2π,2π/100}]

Origin

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

ParametricPlot[{((Cos[t] (8+5 Cos[2 t]+3 Cos[6 t]))/(13+3 Cos[4 t])),((4 (7+6 Cos[2 t]+3 Cos[4 t]) ((Sin[t])^3))/(13+3 Cos[4 t]))},
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic,Ticks->None]

CatacausticPlot[{Cos[t]^3,Sin[t]^3},{0,0},{t,0,2π,2π/100}]

Plot

ParametricPlot[{(Cos[t] (7+6 Cos[t]+4 Cos[2 t]+4 Cos[3 t]+3 Cos[4 t]))/(11+10 Cos[t]+3 Cos[2 t]),((4 (5+4 Cos[t]+3 Cos[2 t]) ((Sin[t])^3))/(11+10 Cos[t]+3 Cos[2 t]))},
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic]

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
CatacausticPlot[{Cos[t]^3,Sin[t]^3},#,{t,0,2π,2π/100}]&/@{
{0,0},
{1,0},
{0,3},
{3,3}
},
ParametricPlot[#,
{t,0,2π},PlotStyle->Blue,AspectRatio->Automatic,Ticks->None]&/@{
{((Cos[t] (8+5 Cos[2 t]+3 Cos[6 t]))/(13+3 Cos[4 t])),((4 (7+6 Cos[2 t]+3 Cos[4 t]) ((Sin[t])^3))/(13+3 Cos[4 t]))},
{(Cos[t] (7+6 Cos[t]+4 Cos[2 t]+4 Cos[3 t]+3 Cos[4 t]))/(11+10 Cos[t]+3 Cos[2 t]),((4 (5+4 Cos[t]+3 Cos[2 t]) ((Sin[t])^3))/(11+10 Cos[t]+3 Cos[2 t]))},
{-(1/2) ((Cos[t])^3) (-5+3 Cos[2 t]),(1/16) (12 Sin[t]-Sin[3 t]+3 Sin[5 t])},
{(1/16) (15 Cos[t]+Cos[3 t]+3 Sin[t]-3 Sin[5 t]),(1/16) (3 Cos[t]-3 Cos[5 t]+15 Sin[t]-Sin[3 t])}
}
}
]]]