Tractrix Spiral

Author

Eric W. Weisstein
April 12, 2021

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

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

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

Source

https://mathcurve.com/courbes2d.gb/spiraletractrice/spiraletractrice.shtml

Plot

With[{a=1},ParametricPlot[{a Cos[t]Cos[Tan[t]-t],a Cos[t]Sin[Tan[t]-t]},{t,0,π},PlotStyle->Red,Ticks->None]]

With[{a=1},ParametricPlot[{a Cos[t]Cos[Tan[t]-t],a Cos[t]Sin[Tan[t]-t]},{t,0,π/2},PlotStyle->Red,Ticks->None]]

Properties

<<MathWorld`Curves`

curve={a Cos[t] Cos[t-Tan[t]],a Cos[t] Sin[t-Tan[t]]};

ArcLength

Assuming[a>0&&0<t0<π/2,ArcLength[curve,{t,0,t0}]]/.t0->t

Curvature

FullSimplify[Curvature[curve,t],a>0&&0<t<π/2]

ImplicitEquation

NormalVector

FullSimplify[NormalVector[curve,t],a>0&&0<t<π/2]/.Abs[x_]^2:>x^2//FullSimplify

ParametricEquations

{a Cos[t]Cos[Tan[t]-t],a Cos[t]Sin[Tan[t]-t]}

Function[t,{a Cos[t]Cos[Tan[t]-t],a Cos[t]Sin[Tan[t]-t]}]/@{0,π/2}

Limit[{a Cos[t]Cos[Tan[t]-t],a Cos[t]Sin[Tan[t]-t]},t->#]&/@{0,π/2}

PolarCurvature

PolarEquation

r==a Cos[t]
θ==Tan[t]-t

Plot[Tan[t]-t,{t,0,π/2}]

Solve[θ==Tan[t]-t,t]

https://mathworld.wolfram.com/Tangent.html

InverseFunction[Tan[#]-#&][t]InverseFunction[Tan[#]-#&][t]

PolarPlot[Cos[InverseFunction[Tan[#]-#&][q]],{q,0,4π},PlotRange->All]

FindInstance[Tan[t]-t==θ,θ]

g[θ_?NumericQ]:=FindRoot[Tan[t]-t==θ,t]

g[.2]

Plot[{Tan[t]-t,InverseFunction[Tan[#]-#&][t],t},{t,0,1.5},AspectRatio->Automatic]

PolarPlot

FullSimplify[SeriesCoefficient[Tan[x],{x,0,n}],n>1]

Sum[(((I^(1+n)) (2^n) (-1+(((-1))^n)) (-1+(2^(1+n))) BernoulliB[1+n])/(Gamma[2+n]))(x^n),{n,∞}]

FullSimplify[%]

Solve[θ==((Sqrt[(a^2)-(r^2)])/r)-ArcCos[(r/a)],r]

PolarSlope

PolarTangentialAngle

Slope

FullSimplify[Slope[curve,t],a>0&&0<t<π/2]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&0<t<π/2]

TangentialAngle

Assuming[0<t<π/2,TangentialAngle[curve,t]]

TangentVector

FullSimplify[TangentVector[curve,t],a>0&&0<t<π/2]/.Abs[x_]^2:>x^2//FullSimplify

VectorLength

FullSimplify[VectorLength[curve,t],a>0&&0<t<π/2]/.Abs[x_]^2:>x^2//FullSimplify