Logarithmic Spiral

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

PolarPlot[Exp[.2t],{t,0,8π},PlotStyle->Red,Ticks->None,PlotRange->All]

With[{a=1,b=.2},
ParametricPlot[{a Cos[t] Exp[b t],a Sin[t]Exp[b t]},{t,0,8Pi},
PlotStyle->Red,PlotRange->All,AspectRatio->Automatic,Ticks->None]
]

Show[Block[{$DisplayFunction=Identity},
Table[With[{a=1},
ParametricPlot[{a Cos[t] Exp[b t],a Sin[t]Exp[b t]},{t,0,8Pi},
PlotStyle->Hue[b]]
],{b,0,1,.01}]],
AspectRatio->Automatic,Ticks->None,PlotRange->20{{-1,1},{-1,1}}
]

Equally Spaced Rays

LogarithmicSpiralPlot[n_:10,its_:10,opts___]:=Module[{
l=Table[{Cos[2Pi i/n],Sin[2Pi i/n]},{i,its}]//N,
spokeopts,spiralopts
},
{spokeopts,spiralopts}={SpokeStyle,SpiralStyle}/.{opts}/.{SpokeStyle->{},SpiralStyle->{}};
{
{spokeopts,
Line[{{0,0},#}]&/@Table[{Cos[2Pi i/n],Sin[2Pi i/n]},{i,n}]//N
},
{spiralopts,
Line[FoldList[Coordinates[EndPoint[PerpendicularLine[Point[#1],Line[{{0,0},#2}]]]]&,{1,0},l]]
}
}
]

Show[Graphics[{LogarithmicSpiralPlot[10,20,SpiralStyle->Red]},AspectRatio->Automatic]]

Show[GraphicsArray[Graphics[#,AspectRatio->Automatic,PlotRange->All]&/@Table[LogarithmicSpiralPlot[n,2n,SpiralStyle->Red],{n,7,11}]]]

Equations

Thread[{x,y}==curve]

Cesàro Equation

Eliminate[{κ==((E^(-b t))/(a (Sqrt[(b^2)+1]))),s==((a (Sqrt[(b^2)+1]) (-1+(E^(b t))))/b)},t]

Properties

LogarithmicSpiral[t_,a_,b_]:={a Cos[t] Exp[b t],a Sin[t]Exp[b t]}

curve=LogarithmicSpiral[t,a,b]

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

Arc Length

From t=0

ArcLength[curve,t,Assumptions->a>0&&b>0&&t>0]

From t=-∞

FullSimplify[Plus@@(D[LogarithmicSpiral[t,a,b],t]^2)]

Integrate[a Sqrt[1+(b^2)] (E^( b t)),{t,-∞,t},Assumptions->b>0]

Integrate[a Sqrt[1+(b^2)] (E^( b t)),{t,0,t},Assumptions->b>0]

Curvature

Parametric

FullSimplify[Curvature[LogarithmicSpiral[t,a,b],t],a>0&&b>0&&t>0]

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

Polar

FullSimplify[Curvature[a Exp[b q],q],a>0&&b>0&&q>0]

Refine[%,a>0&&b>0&&q>0]

Slope

Slope[a Exp[b t] {Cos[t],Sin[t]},t]//FullSimplify

Slope[a Exp[b q],q]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&b>0&&t>0]

Tangential Angle

TangentialAngle[curve,t]

Vector Properties

parm=a Exp[b t] {Cos[t],Sin[t]};
assum=a>0&&b>0&&t>0;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]