Tractrix

Author

Eric W. Weisstein
June 28, 2013

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

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

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

Involute of the Catenary

FullSimplify[Involute[{t,Cosh[t]},t]]

Limit[{t-Tanh[t],Sech[t]},t->0]

Plot

ParametricPlot[{t-Tanh[t],Sech[t]},{t,-4,4},Ticks->None,
AspectRatio->Automatic,
PlotRange->All,PlotStyle->Red,
AxesOrigin->{0,0}]

Equation

tractrix=a{(t-Tanh[t]),Sech[t]};

Solve[{x==a(t-Tanh[t]),y==a Sech[t]},x,t]//FullSimplify

With[{a=1},Plot[-(Sqrt[(a-y) (a+y)])+a ArcSech[(y/a)],{y,-1,1}]]

Solve[{x==a(t-Tanh[t]),y==a Sech[t]},y,t]//FullSimplify

Block[{a=1},ContourPlot[x-a ArcSech[y/a]-Sqrt[a^2-y^2]==0,{x,-2,2},{y,-2,2}]]

Area

Area[tractrix,{t,∞,-∞}]

x[t_]:=a(t-Tanh[t])
y[t_]:=a Sech[t]

Integrate[y[t]x'[t]-x[t]y'[t],{t,-∞,∞}]/2

Properties

curve=a{t-Tanh[t],Sech[t]};

Arc Length

V4.2

ArcLength[{t-Tanh[t],Sech[t]},t]

FullSimplify[Sqrt[(((1-((Sech[t])^2)))^2)+((Sech[t])^2) ((Tanh[t])^2)],t>0]

Integrate[Tanh[t],{t,0,t}]

V5.0

FullSimplify[ArcLength[{t-Tanh[t],Sech[t]},t],t>0]

Log[FullSimplify[Exp[-t-Log[2]+Log[1+Exp[2t]]]]]

Integrate[Tanh[t],{t,0,t}]

V6.1

ArcLength[tractrix,t,Assumptions->a>0&&t>0]

Curvature

V6.1

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

Curvature[

Slope

Slope[a {t-Tanh[t],Sech[t]},t]//FullSimplify

Slope[,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

V4.2

TangentialAngle[{t-Tanh[t],Sech[t]},t]

V5.0

TangentialAngle[{t-Tanh[t],Sech[t]},t]

FullSimplify[2 (-(π/4)+ArcTan[(E^t)])-2 ArcTan[Tanh[(t/2)]],t>0]

Plot[(1/2) (-π+4 ArcTan[(E^t)]-4 ArcTan[Tanh[(t/2)]]),{t,-2,2}]

V6.1

FullSimplify[TangentialAngle[tractrix,t],a>0&&t>0]

Constant-Speed Parametrization

FullSimplify[Integrate[Sqrt[1-Exp[-2t/a]],{t,0,v}]]

FullSimplify[ExpToTrig[%]]

Integrate[Sqrt[1-Exp[-2t/a]],{t,0,v},Assumptions->a>0&&v>0]

FullSimplify[%,a>0&&v>0]

Vector Properties

parm=a {t-Tanh[t],Sech[t]};
assum=a>0&&t∈Reals;
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]]/.xform

NormalVector

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

TangentVector

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