Doppler Spiral

Author

Eric W. Weisstein
April 8, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Parametric equations

GraphicsGrid[Partition[With[{a=1},Table[ParametricPlot[{a(t Cos[t]+k t),a t Sin[t]},{t,0,40},Ticks->None,PlotLabel->Row[{Style["k",Italic]," == ",k}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{k,{0,1/2,1,3/2,2,3}}]],UpTo[3]],Dividers->All]

Properties

ArcLength

Assuming[a>0&&k>0,ArcLength[{a(t Cos[t]+k t),a t Sin[t]},{t,0,t0}]]

FullSimplify[Sqrt[#.#&@D[{a(t Cos[t]+k t),a t Sin[t]}]],a>0&&k≥0&&t>0]

Assuming[k≥0&&t0>0,Integrate[a t (Sqrt[1+(k^2)+2 k Cos[t]]),{t,0,t0}]]//Timing

TrigExpand[Cos[t]]

Integrate[a t (Sqrt[1+(k^2)+2 k Cos[t]]),{t,0,1}]

Integrate[a t (Sqrt[1+(k^2)+2 k Cos[t]]),t]

Curvature

FullSimplify[Curvature[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0]

NormalVector

FullSimplify[NormalVector[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0&&t>0]

ParametricEquations

{a(t Cos[t]+k t),a t Sin[t]}

PolarCurvature

PolarEquation

PolarSlope

PolarTangentialAngle

Slope

FullSimplify[Slope[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0&&t>0]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0&&t>0]

TangentialAngle

TangentialAngle[{a(t Cos[t]+k t),a t Sin[t]},t]

TangentVector

FullSimplify[TangentVector[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0&&t>0]

VectorLength

FullSimplify[VectorLength[{a(t Cos[t]+k t),a t Sin[t]},t],a>0&&k>0&&t>0]