Tschirnhausen Cubic Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Catacaustics for Various Radiant Points

General

cat=FullSimplify[Catacaustic[{3(t^2-3),t(t^2-3)},{x,y},t]]

x=0

FullSimplify[cat/.x->0]

x=-8

FullSimplify[cat/.{x->-8,y->0}]

FullSimplify[Eliminate[{6 (-1+(t^2)),4 (t^3)}=={x,y},t]]

x=∞

FullSimplify[Limit[cat,x->∞]]

y=0

FullSimplify[cat/.y->0]

y=∞

FullSimplify[Limit[cat,y->∞]]

Origin

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

Eliminate[{x,y}=={6 (-13+(t^2)+(1/(1+(t^2)))+(351/(33+(t^2)))),((4 (t^3) (-3+(t^2)) (13+(t^2)))/(33+34 (t^2)+(t^4)))},t]

CatacausticPlot[{3(t^2-3),t(t^2-3)},{-8,0},{t,-3,3,.2}]