Archimedean Spiral

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

PolarPlot[Evaluate[(θ^(1/#))&/@{-2,-1,1,2}],{θ,0,4π},Ticks->None,PlotStyle->{Red,Orange,Green,Blue},ImageSize->300]

PolarPlot[Evaluate[(θ^(1/#))&/@{-2,-1,1,2}],{θ,0,1.2},Ticks->None,PlotStyle->{Red,Orange,Green,Blue},ImageSize->300]

Properties

curve=a t^(1/n){Cos[t],Sin[t]};

Arc Length

ArcLength[curve,t,Assumptions->a>0&&n∈Reals&&t>0]

ArcLength[curve,t,Assumptions->a>0&&n>0&&t>0]//FullSimplify

Doesn't converge for n< 0

curve

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

With[{a=1,n=-1},Integrate[(Sqrt[(((a (t^(1/n)) Cos[t]+((a (t^(-1+(1/n))) Sin[t])/n)))^2)+(((((a (t^(-1+(1/n))) Cos[t])/n)-a (t^(1/n)) Sin[t]))^2)]),{t,0,t}]]

ParametricPlot[{((Cos[t])/t),((Sin[t])/t)},{t,0,10},ImageSize->150]

curve/.n->-1

ArcLength[{((a Cos[t])/t),((a Sin[t])/t)},{t,0,t}]

FullSimplify[(Sqrt[(((((a Cos[t])/t)-((a Sin[t])/(t^2))))^2)+(((-((a Cos[t])/(t^2))-((a Sin[t])/t)))^2)]),a>0&&t>0]

With[{a=1,n=-2},Integrate[(Sqrt[(((a (t^(1/n)) Cos[t]+((a (t^(-1+(1/n))) Sin[t])/n)))^2)+(((((a (t^(-1+(1/n))) Cos[t])/n)-a (t^(1/n)) Sin[t]))^2)]),{t,0,t}]]

ParametricPlot[curve/.{a->1,n->-2},{t,0,10}]

Curvature

FullSimplify[Curvature[curve,t],a>0&&n∈Reals&&t>0]

FullSimplify[Curvature[curve,t],a>0&&n>0&&t>0]

FullSimplify[Curvature[a q^(1/n),q],n>0&&q>0&&a>0]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&n∈Integers]

Slope

FullSimplify[Slope[a t^(1/n) {Cos[t],Sin[t]},t]]

TangentialAngle

TangentialAngle[curve,t,Assumptions->t>0&&n∈Reals]

TangentialAngle[curve,t,Assumptions->t>0&&n>0]

TangentialAngle[curve,t,Assumptions->t>0&&n<0]

Plot[t+ArcTan[2t],{t,0,5}]

Vector Properties

VectorLength

Assuming[a>0&&n>0&&t>0,FullSimplify[Norm[a t^(1/n) {Cos[t],Sin[t]}]]]

NormalVector

Assuming[a>0&&n>0&&t>0,FullSimplify[NormalVector[a t^(1/n) {Cos[t],Sin[t]},t]]]

TangentVector

Assuming[a>0&&n>0&&t>0,FullSimplify[TangentVector[a t^(1/n) {Cos[t],Sin[t]},t]]]