Natural Logarithm Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Curve

ParametricPlot[{t,Log[t]},{t,0,3},PlotStyle->Red]

Analytic

Arbitrary radiant point

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

x=0

FullSimplify[cat/.x->0]

x=∞

FullSimplify[Limit[cat,x->∞]]

ParametricPlot[{((1+(t^2))/(2 t)),-1+Log[t]},{t,.000001,100},
PlotPoints->200,
PlotStyle->Red,PlotRange->{{0,25},{-5,3}}]

FullSimplify[Cosh[Log[t]]]

ParametricPlot[
{{Cosh[u],u-1},
{((1+(u^2))/(2u)),-1+Log[u]}
},{u,-5,5},
PlotStyle->{Red,Blue}]

y=0

FullSimplify[cat/.y->0]

x=∞

FullSimplify[Limit[cat,y->∞]]

ParametricPlot[{{2 t,(1/2)-((t^2)/2)+Log[t]}},{t,.001,3},
PlotStyle->Red]

Plots

CatacausticPlot[{t,Log[t]},{100000,0},{t,.1,10,.2},
PlotRange->{{-1,5},{-4,2}},Axes->True,Ticks->None,
AxesLabel->TraditionalForm/@{x,y}]