Logarithmic Spiral Catacaustic

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Analytic

Arbitrary radiant point

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

x=0

FullSimplify[cat/.x->0]

x=∞

FullSimplify[Limit[cat,x->∞]]

y=0

FullSimplify[cat/.y->0]

y=∞

FullSimplify[Limit[cat,y->∞]]

Origin

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

FullSimplify[Sqrt[#.#]&[{((2 b (E^(b t)) (b Cos[t]-Sin[t]))/(1+(b^2))),((2 b (E^(b t)) (Cos[t]+b Sin[t]))/(1+(b^2)))}],b>0&&t∈Reals]

FullSimplify[Sqrt[#.#]&[{Cos[t]Exp[b t],Sin[t]Exp[b t]}],b>0&&t∈Reals]

Plot

With[{b=1.2},CatacausticPlot[{Cos[t]Exp[b t],Sin[t]Exp[b t]},{0,0},{t,0,6.5,.1},PlotRange->All]
]