Nielsen's Spiral

Author

Eric W. Weisstein
June 28, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Plot

nielsen=a{CosIntegral[t],SinIntegral[t]};

Plot[Evaluate[nielsen],{t,0,10},PlotStyle->{Red,Blue}]

ParametricPlot[nielsen/.a->1,{t,0,50},PlotStyle->Red]

Properties

nielsen=a{CosIntegral[t],SinIntegral[t]};

Arc Length

ArcLength[nielsen,t]

ArcLength[nielsen,{t,t0,t},Assumptions->t>0&&a>0&&t0>0]

ArcLength[nielsen,{t,1,t},Assumptions->t>0&&a>0]

Curvature

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

Slope

Slope[a {CosIntegral[t],SinIntegral[t]},t]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[nielsen,t],a>0&&t>0]

Tangential Angle

TangentialAngle[nielsen,t]

Equations

Polar

FullSimplify[(Sqrt[#.#&[nielsen]]),a>0&&t>0]

PolarPlot[(Sqrt[((CosIntegral[t])^2)+((SinIntegral[t])^2)]),{t,0,10}]

Cesàro

Simplify[Eliminate[{s==a Log[t],κ==t/a},t],a≠0]

Vector Properties

parm=a{CosIntegral[t],SinIntegral[t]};
assum=a>0&&t>0;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

VectorLength

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

NormalVector

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

TangentVector

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