Ellipse Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Catacaustic

(x, y)

cat=FullSimplify[Catacaustic[{a Sin[t],b Cos[t]},{x,y},t]]

(x, 0)

FullSimplify[Limit[cat,x->0]]

(0, y)

FullSimplify[Limit[cat,y->0]]

(0, 0)

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

(∞, y)

FullSimplify[Limit[cat,x->Infinity]]

(x, ∞)

FullSimplify[Limit[cat,y->Infinity]]

Graphics

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{0,0},{t,0,2Pi,.1},PlotRange->{{-1,1},{-1,1}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{.5,0},{t,0,2Pi,.1},PlotRange->{{-2,1},{-1,1}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{9999999,0},{t,0,2Pi,.1},PlotRange->{{-1,1},{-1,1}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{0,9999999},{t,0,2Pi,.1},PlotRange->{{-2,2},{-1,1}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{1.2,0},{t,0,2Pi,.2},PlotRange->{{-2,2},{-1,1}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{0,1},{t,0,2Pi,.2},PlotRange->{{-2,2},{-1,3}},AspectRatio->Automatic]
]

With[{a=1,b=.5},
CatacausticPlot[{a Cos[t],b Sin[t]},{.5,1},{t,0,2Pi,.2},PlotRange->{{-2,2},{-1,2}},AspectRatio->Automatic]
]