Ellipse Evolute

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Equation

FullSimplify[Evolute[{a Cos[t],b Sin[t]},t]]

evolute[a_,b_][t_]:={((((a^2)-(b^2)) ((Cos[t])^3))/a),((((b^2)-(a^2)) ((Sin[t])^3))/b)}

ParametricPlot[evolute[##][t],{t,0,2π}]&@@@{{1,.9},{1,2},{2,1}}

With[{a=1,b=2},ParametricPlot[{((((a^2)-(b^2)) ((Cos[t])^3))/a),((((b^2)-(a^2)) ((Sin[t])^3))/b)},{t,0,2π}]]

With[{a=1,b=2},ParametricPlot[{((-((a^2)-(b^2)) ((Cos[t])^3))/a),((((b^2)-(a^2)) ((Sin[t])^3))/b)},{t,0,2π}]]

With[{a=1,b=2},ParametricPlot[{((((a^2)-(b^2)) ((Cos[t])^3))/a),((-((b^2)-(a^2)) ((Sin[t])^3))/b)},{t,0,2π}]]

Polar

soln=FullSimplify[Solve[Tan[θ]==Simplify[1/Divide@@evolute[a,b][t]],t]]

FullSimplify[FullSimplify[Sqrt[#.#]&[evolute],a>0&&b>0]/.soln[[1]],a>0&&b>0]

FullSimplify[FullSimplify[Sqrt[#.#]&[evolute[a,b][t]],a>0&&b>0]/.soln[[3]],a>0&&b>0]

r[a_,b_][θ_]:=(((a+b) (a-b) (Sqrt[-(((Sec[θ])^2)/(((-1+(((-1))^(1/3)) (((-((b Tan[θ])/a)))^(2/3))))^3))]))/a)

FullSimplify[ExpandAll[-1/(((-1+(((-1))^(1/3)) (((-((b Tan[θ])/a)))^(2/3))))^3)]]

r[a_,b_][θ_]:=(((a+b) (a-b) Abs[Sec[θ]](Sqrt[-1/(((-1+(((-1))^(1/3)) (((-((b Tan[θ])/a)))^(2/3))))^3)]))/a)

PolarPlot[r[##][θ],{θ,0,2π}]&@@@{{1,.9},{1,2},{2,1}}

Cartesian

GroebnerBasis[{x==(((a-b) (a+b) ((Cos[t])^3))/a),y==(((-(a^2)+(b^2)) ((Sin[t])^3))/b),Sin[t]^2+Cos[t]^2==1}/.{Sin[t]->sin,Cos[t]->cos},{x,y},{cos,sin}]

Factor[First[%]]

FullSimplify[%]

AlgebraicDegree[eqn=(-((((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))))^3)+3 (b^2) (((((a^2)-(b^2)))^4)+7 (a^2) ((((a^2)-(b^2)))^2) (x^2)+(a^4) (x^4)) (y^2)-3 (b^4) ((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))) (y^4)+(b^6) (y^6)),{x,y}]

ExpandAll[FullSimplify[(((Expand[a (((a-b) (a+b) ((Cos[t])^3))/a)]))^(2/3))+(((Expand[b (((-(a^2)+(b^2)) ((Sin[t])^3))/b)]))^(2/3)),0<t<Pi/2&&a>b]]

Figure

With[{a=1,b=.5},
Show[GraphicsArray[{EvolutePlot[{a Cos[t],b Sin[t]},{t,0,2π},DisplayFunction->Identity,PlotRange->{{-1.1,1.1},{-1.5,1.5}}],EvolutePlot[{a Cos[t],b Sin[t]},{t,0,2π,.05},4,PlotRange->{{-1.1,1.1},{-1.5,1.5}},DisplayFunction->Identity,AspectRatio->Automatic]}]]
]

Properties

{ParametricPlot[evolute/.{a->1,b->.5},{t,0,π/4}],
ParametricPlot[evolute/.{a->.5,b->1},{t,0,π/4}]}

curve={(a^2-b^2) Cos[t]^3/a,(a^2-b^2) Sin[t]^3/b};

Arc Length Function

ArcLength[evolute,t,Assumptions -> a>0 && b>0 &&0<t<2π]

ArcLength[evolute,{t,0,2π},Assumptions -> a>0 && b>0]

Area

Parametric

Assuming[a>0&&b>0,Area[curve,{t,0,2π}]]//Timing

Inequalities

Area[evolute,{t,0,2π},Assumptions->a>0&&b>0]

With[{a=1,b=.4},RegionPlot[(-((((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))))^3)+3 (b^2) (((((a^2)-(b^2)))^4)+7 (a^2) ((((a^2)-(b^2)))^2) (x^2)+(a^4) (x^4)) (y^2)-3 (b^4) ((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))) (y^4)+(b^6) (y^6))<0,{x,-1,1},{y,-2.1,2.1},MaxRecursion->5]]

Assuming[a>0&&b>0&&a≠b,Area[(-((((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))))^3)+3 (b^2) (((((a^2)-(b^2)))^4)+7 (a^2) ((((a^2)-(b^2)))^2) (x^2)+(a^4) (x^4)) (y^2)-3 (b^4) ((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))) (y^4)+(b^6) (y^6))<0,{x,y}]]

FullSimplify[%,a>0&&b>0&&a≠b]

Area Inertia Tensor

Parametric

Assuming[a>0&&b>0,AreaInertiaTensor[curve,{t,0,2π}]]//Timing

Centroid

Parametric

Assuming[a>0&&b>0,Centroid[curve,{t,0,2π}]]//Timing

Inequalities

Assuming[a>0&&b>0&&a≠b,Centroid[(-((((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))))^3)+3 (b^2) (((((a^2)-(b^2)))^4)+7 (a^2) ((((a^2)-(b^2)))^2) (x^2)+(a^4) (x^4)) (y^2)-3 (b^4) ((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))) (y^4)+(b^6) (y^6))<0,{x,y}]]

Assuming[a>0&&b>0&&a≠b,Centroid[(-((((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))))^3)+3 (b^2) (((((a^2)-(b^2)))^4)+7 (a^2) ((((a^2)-(b^2)))^2) (x^2)+(a^4) (x^4)) (y^2)-3 (b^4) ((a^4)+(b^4)-(a^2) (2 (b^2)+(x^2))) (y^4)+(b^6) (y^6))<0,{y,x}]]

Curvature

FullSimplify[Curvature[evolute,t],a>0&&b>0&&0<t<2π]

Curvature[((a^12)-6 (a^10) (b^2)+15 (a^8) (b^4)-20 (a^6) (b^6)+15 (a^4) (b^8)-6 (a^2) (b^10)+(b^12)-3 (a^10) (x^2)+12 (a^8) (b^2) (x^2)-18 (a^6) (b^4) (x^2)+12 (a^4) (b^6) (x^2)-3 (a^2) (b^8) (x^2)+3 (a^8) (x^4)-6 (a^6) (b^2) (x^4)+3 (a^4) (b^4) (x^4)-(a^6) (x^6)-3 (a^8) (b^2) (y^2)+12 (a^6) (b^4) (y^2)-18 (a^4) (b^6) (y^2)+12 (a^2) (b^8) (y^2)-3 (b^10) (y^2)-21 (a^6) (b^2) (x^2) (y^2)+42 (a^4) (b^4) (x^2) (y^2)-21 (a^2) (b^6) (x^2) (y^2)-3 (a^4) (b^2) (x^4) (y^2)+3 (a^4) (b^4) (y^4)-6 (a^2) (b^6) (y^4)+3 (b^8) (y^4)-3 (a^2) (b^4) (x^2) (y^4)-(b^6) (y^6)),{x,y}]

SortBy[Simplify[%,a>0&&b>0&&(x|y)∈Reals],ByteCount]

ByteCount/@%

SortBy[FullSimplify[%%,a>0&&b>0&&(X|y)∈Reals],ByteCount]

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]

Singular Points

FullSimplify[SingularPoints[eqn,{x,y}],a>0&&b>0]//Union

Slope

Slope[{(a^2-b^2) Cos[t]^3/a,(-a^2+b^2) Sin[t]^3/b},t]//FullSimplify

Slope[(a^4+b^4-a^2 (2 b^2+x^2))^3+3 b^2 ((a^2-b^2)^4+7 a^2 (a^2-b^2)^2 x^2+a^4 x^4) y^2-3 b^4 (a^4+b^4-a^2 (2 b^2+x^2)) y^4+b^6 y^6==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

TangentialAngle

FullSimplify[TangentialAngle[evolute,t],a>0&&b>0&&0<t<2π]

FullSimplify[D[(1/(4 a b (-(a^2)+(b^2))))Abs[(a^2)-(b^2)] (4 (b^3)-(Sqrt[2]) ((Csc[2 t])^3) (((((a^2)+(b^2)+(-(a^2)+(b^2)) Cos[2 t]) ((Sin[2 t])^2)))^(3/2))),t](((a^2) (b^2))/(3 Abs[(a-b) (a+b) Cos[t] Sin[t]] ((((b^2) ((Cos[t])^2)+(a^2) ((Sin[t])^2)))^(3/2)))),a>0&&b>0&&t>0]

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

Vector Properties

parm=(a^2-b^2) {Cos[t]^3/a,Sin[t]^3/b};
assum=a>0&&b>0&&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]]