Ellipse Involute

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Equation

involute=FullSimplify[Involute[{a Cos[t],b Sin[t]},t],a>0&&b>0]

{x,y}={a (Cos[t]+(((Sqrt[2]) b EllipticE[t,1-((a^2)/(b^2))] Sin[t])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))),b (-(((Sqrt[2]) b Cos[t] EllipticE[t,1-((a^2)/(b^2))])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))+Sin[t])};

Simplify[b Cos[t] x-a Sin[t]y]

Plot

With[{a=1,b=.7},
InvolutePlot[{a Cos[t],b Sin[t]},{t,0,4π},PlotRange->All]
]

Curve Properties

curve={a (Cos[t]+(((Sqrt[2]) b EllipticE[t,1-((a^2)/(b^2))] Sin[t])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))),b (-(((Sqrt[2]) b Cos[t] EllipticE[t,1-((a^2)/(b^2))])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))+Sin[t])};

Arc Length Function

Simplify[(Sqrt[Total[Simplify[D[involute,t]^2]]]),a>0&&b>0&&t>0]

Integrate[((2 a (b^2) EllipticE[t,1-((a^2)/(b^2))])/((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t])),{t,0,t}]

Curvature

Simplify[Together//@Curvature[involute,t]]

FullSimplify[%,a>0&&b>0&&t>0]

FullSimplify[Abs[EllipticE[t,1-x^2]],t>0&&x∈Reals]

Table[Abs[EllipticE[t,1-(x^2)]]-EllipticE[t,1-(x^2)]/.{t->RandomReal[{0,10}],x->RandomReal[{-10,10}]},{1000}]//Chop//Union

Slope

FullSimplify[Slope[{a (Cos[t]+(Sqrt[2] b EllipticE[t,1-a^2/b^2] Sin[t])/Sqrt[a^2+b^2+(-a^2+b^2) Cos[2 t]]),b (Sin[t]-(Sqrt[2] b EllipticE[t,1-a^2/b^2] Cos[t])/Sqrt[a^2+b^2+(-a^2+b^2) Cos[2 t]])},t],a>0&&b>0]

SpecialAffineCurvature

ByteCount[k=SpecialAffineCurvature[{a (Cos[t]+(((Sqrt[2]) b EllipticE[t,1-((a^2)/(b^2))] Sin[t])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))),b (-(((Sqrt[2]) b Cos[t] EllipticE[t,1-((a^2)/(b^2))])/(Sqrt[(a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]]))+Sin[t])},t]]//Timing

ByteCount[k2=FullSimplify[k,a>0&&b>0&&t>0]]//Timing

ran overnight

Plot[Sin[2t],{t,0,2π}]

k2

TangentialAngle

FullSimplify[TangentialAngle[involute,t],a>0&&b>0&&t>0

FullSimplify[((2 a (b^2) EllipticE[t,1-((a^2)/(b^2))])/((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]))(1/(b EllipticE[t,1-((a^2)/(b^2))]))]

Integrate[((2 a b)/((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t])),{t,0,t}]

Vector Properties

parm={a (Cos[t]+(Sqrt[2] b EllipticE[t,1-a^2/b^2] Sin[t])/Sqrt[a^2+b^2+(-a^2+b^2) Cos[2 t]]),b (Sin[t]-(Sqrt[2] b EllipticE[t,1-a^2/b^2] Cos[t])/Sqrt[a^2+b^2+(-a^2+b^2) Cos[2 t]])};
assum=a>0&&b>0&&t>0;
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]]