Parabola Involute

Author

Eric W. Weisstein
June 28, 2013

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

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

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

Curve

i=FullSimplify[Involute[{a t^2,a t},t],t>0]

TrigReduce[ArcSinh[2t]]

Solve[u==ArcSinh[2t],t]

FullSimplify[i/.t->Sinh[u]/2,u>0]

Plot

With[{a=1},ParametricPlot[{{t^2,t},{-((a t ArcSinh[2 t])/(2 (Sqrt[1+4 (t^2)]))),(1/4) a (2 t-((ArcSinh[2 t])/(Sqrt[1+4 (t^2)])))}},{t,-3,3},
PlotStyle->{{},Red},
AspectRatio->Automatic,Ticks->None
]]

With[{a=1},ParametricPlot[{{t^2,t},{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])}},{t,-3,3},
PlotStyle->{{},Red},
AspectRatio->Automatic,Ticks->None
]]

With[{a=1},ParametricPlot[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},{t,-2,2},
PlotStyle->Red,
AspectRatio->Automatic
]]

Properties

ArcLengthFunction

ArcLength[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t]

FullSimplify[%,a>0&&t∈Reals]

s=(1/4) (-1+2 Catalan+(Abs[t Sech[t]+Sinh[t]] Cosh[t] (-2 t ArcCot[(E^t)]+Cosh[t]-I PolyLog[2,-I (E^(-t))]+I PolyLog[2,I (E^(-t))]))/(t+Cosh[t] Sinh[t]));

Limit[s,t->#]&/@{-∞,0,∞}

Plot[(1/4) (-1+2 Catalan+(Abs[t Sech[t]+Sinh[t]] Cosh[t] (-2 t ArcCot[(E^t)]+Cosh[t]-I PolyLog[2,-I (E^(-t))]+I PolyLog[2,I (E^(-t))]))/(t+Cosh[t] Sinh[t])),{t,-1,1}]

Curvature

FullSimplify[Curvature[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t],a>0&&t∈Reals]

FullSimplify[SpecialAffineCurvature[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t],a>0&&t∈Reals]

Slope

FullSimplify[Slope[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t],t∈Reals]

TangentialAngle

FullSimplify[TangentialAngle[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t],t∈Reals]

VectorLength

FullSimplify[VectorLength[{-(1/4) a t Tanh[t],(1/4) a (-t Sech[t]+Sinh[t])},t],t∈Reals&&a>0]

Vector Properties

parm={-(1/4) a t Tanh[t],1/4 a (-t Sech[t]+Sinh[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),
t Sech[t]+Sinh[t]≥0:>t≥0,t Sech[t]+Sinh[t]<0:>t<0};

parm/.t->0

VectorLength

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

NormalVector

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

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

FullSimplify[Subtract@@{Piecewise[{{-Tanh[t], t >= 0}}, Tanh[t]],-Sign[t]Tanh[t]}]

FullSimplify[Subtract@@{Piecewise[{{-Sech[t], t >= 0}}, Sech[t]],-Sign[t]Sech[t]}]

FullSimplify[Subtract@@{Piecewise[{{-Sech[t], t >= 0}}, Sech[t]],-Sign[t]Sech[t]}]

{Piecewise[{{-Sech[t], t >= 0}}, Sech[t]],-Sign[t]Sech[t]}/.t->0

TangentVector

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

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

FullSimplify[t Sech[t]+Sinh[t]≥0,t∈Reals]

Plot[{t Sech[t],Sinh[t],t Sech[t]+Sinh[t]},{t,-1,1}]