Archimedes' Spiral

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

Polar

PolarPlot[t,{t,0,6Pi},Ticks->None,PlotStyle->Red]

Cartesian

ParametricPlot[t{Cos[t],Sin[t]},{t,0,6Pi},Ticks->None,PlotStyle->Red,ImageSize->150]

Properties

spiral=a t{Cos[t],Sin[t]};

Arc Length

Cartesian

FullSimplify[ArcLength[spiral,t],t>0]

Polar

PolarArcLength[a θ,θ,Assumptions->θ>0&&a>0]

Simplify[TrigToExp[%]]

%/.θ->0

<<DiscreteMath`RSolve`

Series[(1/2) (θ (Sqrt[1+(θ^2)])+ArcSinh[θ]),{θ,0,10}]

s=SeriesTerm[(1/2) (θ (Sqrt[1+(θ^2)])+ArcSinh[θ]),{θ,0,n}]

FullSimplify[s,1<=n≤2]

FullSimplify[s,n>=3]

Table[(1/2) (LegendreP[-3+n,0]+(((1+n) LegendreP[-1+n,0])/n)),{n,3,7}]

Last/@Partition[CoefficientList[Normal[Series[(1/2) (θ (Sqrt[1+(θ^2)])+ArcSinh[θ]),{θ,0,30}]],θ],2]

Through[{Numerator,Denominator}[%]]

Curvature

FullSimplify[Curvature[a q,q],a>0]

FullSimplify[Curvature[spiral,t],a>0&&t>0]

FullSimplify[SpecialAffineCurvature[spiral,t],a>0]

Slope

FullSimplify[Slope[a t{Cos[t],Sin[t]},t]]

Slope[a q,q]//FullSimplify

Tangential Angle

TangentialAngle[spiral,t,Assumptions->a>0&&t>0]

Vector Properties

VectorLength

Assuming[a>0&&t>0,FullSimplify[Norm[a t{Cos[t],Sin[t]}]]]

NormalVector

Assuming[a>0&&t>0,FullSimplify[NormalVector[a t{Cos[t],Sin[t]},t]]]

TangentVector

Assuming[a>0&&t>0,FullSimplify[TangentVector[a t{Cos[t],Sin[t]},t]]]