Logarithmic Spiral Evolute

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equation

curve=FullSimplify[Evolute[{Cos[t],Sin[t]}a E^(b t),t]]

Graphic

movie=Show/@With[{a=1,b=.5},
EvoluteFrames[{-a b (E^(b t)) Sin[t],a b (E^(b t)) Cos[t]},{t,0,5,.1},LineLength->5,PlotRange->{{-1,1},{-1,1}}2,
AspectRatio->Automatic]
]

Show[Graphics[movie[[-1]]]]

<<Utilities`GIF`;

Export[gifs<>"logspev.gif",movie]//Timing

Properties

Arc Length

FullSimplify[ArcLength[curve,t]]

FullSimplify[%,a>0&&b>0&&t>0]

Curvature

FullSimplify[Curvature[curve,t]]

FullSimplify[%,a>0&&b>0&&t>0]

LeafCount[%]

PowerExpand[%42]

LeafCount[%]

v5.0

Refine[(1/(Sqrt[(a^2) (b^2) (1+(b^2)) (E^(2 b t))])),a>0&&b>0&&t>0]

Tangential Angle

FullSimplify[TangentialAngle[curve,t]]