Circle Involute

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Curve

circleinvolute=FullSimplify[Involute[a{Cos[t],Sin[t]},t]]

circleinvolute={a (Cos[t]+t Sin[t]),a (-t Cos[t]+Sin[t])};

Plot

ParametricPlot[{{Cos[t]+t Sin[t],Sin[t]-t Cos[t]},{Cos[t/1.75],Sin[t/1.75]}},{t,0,3.5 Pi},Ticks->None,AspectRatio->Automatic,PlotStyle->{Red,Black},PlotRange->{{-10,8},{-7,10}}]

ParametricPlot[{{Cos[t]+t Sin[t],Sin[t]-t Cos[t]},{Cos[2t/1.5],Sin[2t/1.5]}},{t,0,1.5 Pi},Ticks->None,AspectRatio->Automatic,PlotStyle->{Red,Black}]

Properties

circleinvolute={a (Cos[t]+t Sin[t]),a (-t Cos[t]+Sin[t])};

Arc Length Function

v5

Simplify[ArcLength[circleinvolute,t],t>0&&a>0]

v6.1

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

Curvature

v5

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

v6

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

Slope

Slope[a{Cos[t]+t Sin[t],Sin[t]-t Cos[t]},t]//FullSimplify

SpecialAffineCurvature

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

Tangental Angle

v5

TangentialAngle[circleinvolute,t]

v6.1

TangentialAngle[circleinvolute,t,Assumptions->a>0&&t>0]

Vector Properties

parm=(a+k) {Cos[t],Sin[t]};
assum=a>0&&k∈Reals&&0<t<2π;
xform=Abs[x_]^(n_/;Mod[n,2]==0):>x^n;

VectorLength

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

NormalVector

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

TangentVector

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