Talbot's Curve

Author

Eric W. Weisstein
June 28, 2013

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

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

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

Derivation

np=FullSimplify[NegativePedalCurve[{a Cos[t],b Sin[t]},{0,0},t]]

eq=Simplify[np/.Cos[t]^2->(1-Sin[t]^2)/.Cos[t]^3->Cos[t](1-Sin[t]^2)]

Equation

talbot[a_,b_][t_]:=Module[{c=Sqrt[a^2-b^2]},
{(a^2+c^2Sin[t]^2)Cos[t]/a,(a^2-2c^2+c^2Sin[t]^2)Sin[t]/b}
]

talbot[{a,b},t]

FullSimplify[%]

talbot[{a,b},t]//TraditionalForm

Polar

FullSimplify[1/Divide@@talbot[a,b][t]]

Solve[Tan[θ]==((a (-(a^2)+(b^2)) Cos[t] Sin[t]+a (b^2) Tan[t])/(b ((a^2)+(a-b) (a+b) ((Sin[t])^2)))),t]//Simplify

Refine[FullSimplify[Sqrt[#.#]&[talbot[a,b][t]],a>0&&b>0],a>0&&b>0]

PolarPlot[Function[{a,b},Function[t,(1/(2 (Sqrt[2]) a b))(√(-(-(a^2)-(b^2)+(a-b) (a+b) Cos[2 t]) ((((a^2)+(b^2)+(a-b) (a+b) Cos[2 t]))^2)))]][##][t],{t,0,2π}]&@@@{{3,1},{2,1},{1.5,1},{1,1},{1,1.5},{1,2}}

Cartesian

Numerator/@Factor/@GroebnerBasis[Append[Thread[{x,y}==talbot[{a,b},t]],Cos[t]^2+Sin[t]^2==1],{x,y},{Sin[t],Cos[t]},MonomialOrder->EliminationOrder]//Union

eqn=-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6);

AlgebraicDegree[eqn,{x,y}]

With[{a=1,b=.45},ContourPlot[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)==0,{x,-1.1,1.1},{y,-1,1},PlotPoints->70]]

With[{a=1,b=.45},RegionPlot[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0,{x,-1.1,1.1},{y,-1,1},PlotPoints->70]]

Shape

Plot

ParametricPlot[Evaluate[talbot[{1,.45},t]],{t,0,2Pi},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

ParametricPlot[Evaluate[talbot[{1,.45},t]],{t,0,Pi},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

ParametricPlot[talbot[##][t],{t,0,2π}]&@@@{{3,1},{2,1},{1.5,1},{1,1},{1,1.5},{1,2}}

Sqrt[2.]

Manipulate[ParametricPlot[talbot[a,b][t],{t,0,2π}],{{a,1.5},.1,2},{{b,1},.1,2}]

Sqrt[2.]

Manipulate[ParametricPlot[talbot[a,b][t],{t,0,2π},PlotRange->{{1.4,1.5},{-.1,.1}.5},AspectRatio->1],{{a,1.5},.1,2},{{b,1},.1,2}]

With[{a=1,b=.45},RegionPlot[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0,{x,-1.2,1.2},{y,-.5,.5},PlotPoints->80,AspectRatio->Automatic,Epilog->{Red,Point[{{-((2 b (Sqrt[(a-b) (a+b)]))/a),0},{((2 b (Sqrt[(a-b) (a+b)]))/a),0}}]}]]

With[{a=Sqrt[2],b=1},RegionPlot[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0,{x,-2,2},{y,-1,1},PlotPoints->80,AspectRatio->Automatic,Epilog->{Red,Point[{{-a,0},{a,0}}]}]]

With[{a=.45,b=1},RegionPlot[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0,{x,-.5,.5},{y,-1.2,1.2},PlotPoints->80,AspectRatio->Automatic,Epilog->{Red,Point[{{0,-((2 a (Sqrt[-(a-b) (a+b)]))/b)},{0,((2 a(Sqrt[-(a-b) (a+b)]))/b)}}]}]]

Limit[((2 a (Sqrt[(a-b) (a+b)]))/b),a->Sqrt[2]b]

Tails

talbot[a,b][t]

t/.Assuming[a>0&&b>0&&0<t<2π,Solve[((Sin[t] ((a^2)-2 ((a^2)-(b^2))+((a^2)-(b^2)) ((Sin[t])^2)))/b)==0,t]]

Limit[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)])),a->b,Direction->1]

With[{a=1,b=.45},ParametricPlot[talbot[a,b][t],{t,-ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))],ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))]}]]

With[{a=1,b=.45},ParametricPlot[talbot[a,b][t],{t,0,ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))]}]]

With[{a=1,b=.45},ParametricPlot[talbot[a,b][t],{t,ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))],π/2}]]

With[{a=1,b=.45},ParametricPlot[talbot[a,b][t],{t,ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))],π-ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))]}]]

Special Cases

Circle

talbot[a,a][t]

Parallel Curve

p=FullSimplify[ParallelCurve[{a Cos[t],b Sin[t]},t,d]]

Simplify[p/.Cos[t]:>Sqrt[1-Sin[t]^2]]

With[{a=1,b=.5},
ParallelCurvesPlot[{a Cos[t],b Sin[t]},{-.43,-.43,.1},{t,0,2Pi}]
]

With[{a=1,b=.5},
ParametricPlot[
Evaluate[Join[
{ParallelCurve[{a Cos[t],b Sin[t]},t,-.4]},
Table[
ParallelCurve[{a Cos[t],b Sin[t]},t,k],{k,-1,0,.1}]]
],{t,0,2Pi},
PlotStyle->Prepend[Table[Hue[h],{h,0,1,.1}],{Thickness[.012],Black}],
Ticks->None]
]

With[{a=1,b=.5},
ParallelCurvesPlot[{a Cos[t],b Sin[t]},{-1,0,.1},{t,0,2Pi}]
]

Properties

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

ArcLength Function

FullSimplify[Sqrt[#.#&[D[talbot[a,b][t],t]]],1/(Sqrt[2]) < a/b<1&&0<t<2π]

V5

ArcLength[talbot[{a,b},t],t,Assumptions->1/(Sqrt[2]) < a/b<1&&0<t<2π]

V6

Integrate[(((Sqrt[(a^2)+(b^2)+(a-b) (a+b) Cos[2 t]]) ((a^2)+(b^2)+3 (-(a^2)+(b^2)) Cos[2 t]))/(2 (Sqrt[2]) a b)),t]

f[a_,b_][t_]:=(2 (Sqrt[2]) (a^2) (b^2) (Sqrt[(((a^2)+(b^2)+((a^2)-(b^2)) Cos[2 t])/(a^2))]) EllipticF[t,1-((b^2)/(a^2))]-((a^2)-(b^2)) ((a^2)+(b^2)+((a^2)-(b^2)) Cos[2 t]) Sin[2 t])/(2 (Sqrt[2]) a b (Sqrt[(a^2)+(b^2)+((a^2)-(b^2)) Cos[2 t]]))

1/(Sqrt[2]) < a/b<1//N

Plot[f[.707,1][t],{t,0,2π}]

ArcLength

FullSimplify[ArcLength[talbot[{a,b},t],{t,0,2π},Assumptions->1/(Sqrt[2]) < a/b<1]]

With[{a=1,b=.451},{a EllipticK[1-((a^2)/(b^2))],b EllipticK[1-((b^2)/(a^2))]}]

4a EllipticK[1-((a^2)/(b^2))]

Area

area[a_,b_]:=Piecewise[{{(2 Sqrt[Abs[a^2-b^2]] Sqrt[Max[a,b]^2-2 Min[a,b]^2] Sqrt[Min[a,b]^2/Abs[a^2-b^2]] (Max[a,b]^2+13 Min[a,b]^2)+1/2 (a^4-10 a^2 b^2+b^4) (Pi-4 ArcCos[Sqrt[Max[a,b]^2-2 Min[a,b]^2]/Sqrt[Abs[a^2-b^2]]]))/(4 a b),0<Sqrt[2] Min[a,b]<Max[a,b]},{-(a^4-10 a^2 b^2+b^4) Pi/(8 a b),0<Max[a,b]<Sqrt[2] Min[a,b]}}]

area[a,b]//TraditionalForm

Tail piece for a>(Sqrt[2])b

{f,g}=talbot[a,b][t]

-g D[f,t]//FullSimplify

With[{a=1,b=.45},NIntegrate[Boole[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0],{x,((2 b (Sqrt[(a-b) (a+b)]))/a),∞},{y,-∞,∞}]]

part1=-2Integrate[((((a^2)-3 (b^2)+(a-b) (a+b) Cos[2 t]) (-(a^2)-(b^2)+3 (a-b) (a+b) Cos[2 t]) ((Sin[t])^2))/(4 a b)),{t,0,ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))]}]//FullSimplify

%/.{a->1,b->.45}

Middle part for a>(Sqrt[2])b

With[{a=1,b=.45},2NIntegrate[Boole[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0],{x,0,((2 b (Sqrt[(a-b) (a+b)]))/a)},{y,-∞,∞}]]

part2=4Integrate[((((a^2)-3 (b^2)+(a-b) (a+b) Cos[2 t]) (-(a^2)-(b^2)+3 (a-b) (a+b) Cos[2 t]) ((Sin[t])^2))/(4 a b)),{t,ArcSin[((Sqrt[(a^2)-2 (b^2)])/(Sqrt[(a^2)-(b^2)]))],π/2}]//FullSimplify

part2/.{a->1,b->.45}

Both for a>(Sqrt[2])b

area1=FullSimplify[2part1+part2]

area1/.{a->1,b->.45}

With[{a=1,b=.45},NIntegrate[Boole[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0],{x,-∞,∞},{y,-∞,∞}]]

area[1,.45]

area[.45,1]

Whole curve for b<a<(Sqrt[2])b

area2=Integrate[((((a^2)-3 (b^2)+(a-b) (a+b) Cos[2 t]) (-(a^2)-(b^2)+3 (a-b) (a+b) Cos[2 t]) ((Sin[t])^2))/(4 a b)),{t,0,2π}]//FullSimplify

area2/.{a->1.1,b->1}

With[{a=1.1,b=1},NIntegrate[Boole[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0],{x,-∞,∞},{y,-∞,∞}]]

area[1.1,1]

area[1,1.1]

area[1,1]

AreaInertiaTensor

Parametric

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

FullSimplify[Last[%]]

BoundingBox

maxx=Maximize[{x,-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)==0},{x,y}]

maxy=Maximize[{y,-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)==0},{x,y}]

FullSimplify[maxx,0<a<(Sqrt[2])b]

FullSimplify[{maxx[[1]],maxy[[1]]},0<a<(Sqrt[2])b]

FullSimplify[{maxx[[1]],maxy[[1]]},0<(Sqrt[2])b<a]

FullSimplify[(2/(3 (Sqrt[3]) (Sqrt[(((a^2) (a-b) (a+b))/(((2 (a^2)-(b^2)))^3))])))-((2 (((2 (a^2)-(b^2)))^(3/2)))/(3 (Sqrt[3]) (Sqrt[(a^2) (a-b) (a+b)]))),a>0&&b>0]

{((2 (((2 (a^2)-(b^2)))^(3/2)))/(3 (Sqrt[3]) (Sqrt[(a^2) (a-b) (a+b)]))),((2 ((((a^2)-2 (b^2)))^(3/2)))/(3 (Sqrt[3]) b (Sqrt[(a-b) (a+b)])))}/.{a->3.,b->1.}

Centroid

Parametric

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

Inequality

Assuming[a>0&&b>0,Centroid[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0,{x,y}]]//Timing

Curvature

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

FullSimplify[Curvature[talbot[{a,b},t],t],1/(Sqrt[2]) < a/b<1&&0<t<2π]

Implicit Curvature

FullSimplify[ImplicitCurvature[eqn,{x,y}]]

Inflection Points

InflectionPoints[eqn,{x,y}]

SignedArea

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

area/.b->a

{f,g}=talbot[a,b][t]

FullSimplify[-g D[f,t],a>0&&b>0]

Assuming[{a>0&&b>0},FullSimplify@Integrate[((((a^2)-3 (b^2)+(a-b) (a+b) Cos[2 t]) (-(a^2)-(b^2)+3 (a-b) (a+b) Cos[2 t]) ((Sin[t])^2))/(4 a b)),{t,0,2π}]]

-((((a^4)-10 (a^2) (b^2)+(b^4)) π)/(8 a b))/.{a->1,b->.45}

With[{a=1,b=.45},NIntegrate[Boole[-16 (a^8) (b^4)+32 (a^6) (b^6)-16 (a^4) (b^8)+8 (a^8) (b^2) (x^2)+8 (a^6) (b^4) (x^2)-32 (a^4) (b^6) (x^2)+16 (a^2) (b^8) (x^2)-(a^8) (x^4)-8 (a^6) (b^2) (x^4)+8 (a^4) (b^4) (x^4)+(a^6) (x^6)+16 (a^8) (b^2) (y^2)-32 (a^6) (b^4) (y^2)+8 (a^4) (b^6) (y^2)+8 (a^2) (b^8) (y^2)-20 (a^6) (b^2) (x^2) (y^2)+38 (a^4) (b^4) (x^2) (y^2)-20 (a^2) (b^6) (x^2) (y^2)+3 (a^4) (b^2) (x^4) (y^2)+8 (a^4) (b^4) (y^4)-8 (a^2) (b^6) (y^4)-(b^8) (y^4)+3 (a^2) (b^4) (x^2) (y^4)+(b^6) (y^6)<0],{x,-1.1,1.1},{y,-1,1}]]

Singular Points

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

Slope

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

Slope[-16 a^8 b^4+32 a^6 b^6-16 a^4 b^8+8 a^8 b^2 x^2+8 a^6 b^4 x^2-32 a^4 b^6 x^2+16 a^2 b^8 x^2-a^8 x^4-8 a^6 b^2 x^4+8 a^4 b^4 x^4+a^6 x^6+16 a^8 b^2 y^2-32 a^6 b^4 y^2+8 a^4 b^6 y^2+8 a^2 b^8 y^2-20 a^6 b^2 x^2 y^2+38 a^4 b^4 x^2 y^2-20 a^2 b^6 x^2 y^2+3 a^4 b^2 x^4 y^2+8 a^4 b^4 y^4-8 a^2 b^6 y^4-b^8 y^4+3 a^2 b^4 x^2 y^4+b^6 y^6==0,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&b>0&&0<t<2π]

Tangential Angle

V5

TangentialAngle[talbot[{a,b},t],t,Assumptions->0<t<2π]

TangentialAngle[talbot[{a,b},t],t,Assumptions->0<t<π/2]

V6

TangentialAngle[talbot[a,b][t],t,Assumptions->0<t<2π]//Timing

TangentialAngle[talbot[a,b][t],t,Assumptions->0<t<π/2]//Timing

TangentialAngle[talbot[a,b][t],t,Assumptions->π/2<t<π]//Timing

With[{b=1,a=.8},Plot[ArcTan[((b Tan[t])/a)],{t,0,2π}]]

With[{b=1,a=.71},Plot[ArcTan[((b Tan[t])/a)]+Floor[t/π+1/2]π,{t,0,2π}]]

Vector Properties

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