Parabola Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equations

General

cat=FullSimplify[Catacaustic[{t,t^2},{x,y},t]]

x=0

FullSimplify[cat/.x->0]

x=∞

FullSimplify[Limit[cat,x->∞]]

Tschirnhausen cubic with a=9/4

FullSimplify[{((3 t)/2)-2 (t^3),3 (t^2)}/.t->3/2t]

y=0

FullSimplify[cat/.y->0]

y=∞, a focus

Limit[cat,y->∞]

Plots

CatacausticPlot[{t,t^2},{0,10000},{t,-1,1,.1},
PlotRange->{{-1.,1.},{-1,2}}]

CatacausticPlot[{t,t^2},{10000,0},{t,-1,1,.05},
PlotRange->{{-2.,2.},{-1,4}}]