Bifoliate

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Equations

eqn=x^4+y^4-2a x y^2;

FullSimplify[x^4+y^4==2a x y^2/.Thread[{x,y}->r{Cos[θ],Sin[θ]}]]

Solve[(r (3+Cos[4 θ])-8 a Cos[θ] ((Sin[θ])^2))==0,r]

FullSimplify[((8 a Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])),0<θ<π]

Degree

AlgebraicDegree[x^4+y^4==2 x y^2,{x,y}]

Plot

Implicit

<<Graphics`ImplicitPlot`

ImplicitPlot[x^4+y^4==2 x y^2,{x,-3,3},
Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

Polar

PolarPlot[((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])),{θ,0,#},Ticks->None,PlotStyle->Red,ImageSize->Small]&/@{π/2,π}

Parametric

curve=((8a Cos[t] ((Sin[t])^2))/(3+Cos[4 t])){Cos[t],Sin[t]};

ParametricPlot[curve/.a->1,{t,0,π},ImageSize->Small]

Properties

curve=(8 a Cos[t] Sin[t]^2)/(3+Cos[4 t]) {Cos[t],Sin[t]};

Arc Length

V5?

(s=ArcLength[((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])),θ])//Timing

FullSimplify[√(((64 ((Cos[θ])^2) ((Sin[θ])^4))/(((3+Cos[4 θ]))^2))+(((((16 ((Cos[θ])^2) Sin[θ])/(3+Cos[4 θ]))-((8 ((Sin[θ])^3))/(3+Cos[4 θ]))+((32 Cos[θ] ((Sin[θ])^2) Sin[4 θ])/(((3+Cos[4 θ]))^2))))^2)),0<θ<π/2]

(s=2NIntegrate[(1/(((3+Cos[4 θ]))^2))2 √(364+348 Cos[2 θ]+304 Cos[4 θ]+38 Cos[6 θ]-28 Cos[8 θ]-2 Cos[10 θ]) Sin[θ],{θ,0,π/2},WorkingPrecision->150,MaxRecursion->10])//Timing

s[t_?NumericQ]:=2NIntegrate[(1/(((3+Cos[4 θ]))^2))2 (Sqrt[364+348 Cos[2 θ]+304 Cos[4 θ]+38 Cos[6 θ]-28 Cos[8 θ]-2 Cos[10 θ]]) Sin[θ],{θ,0,t}]

Plot[s[t],{t,0,π},PlotStyle->Red]

2Integrate[(1/(((3+Cos[4 θ]))^2))2 √(364+348 Cos[2 θ]+304 Cos[4 θ]+38 Cos[6 θ]-28 Cos[8 θ]-2 Cos[10 θ]) Sin[θ],{θ,0,π/2}]//Timing

V6.1

ArcLength[curve,{t,0,π},Assumptions->a>0]

FullSimplify[√((((((16 a ((Cos[t])^3) Sin[t])/(3+Cos[4 t]))-((16 a Cos[t] ((Sin[t])^3))/(3+Cos[4 t]))+((32 a ((Cos[t])^2) ((Sin[t])^2) Sin[4 t])/(((3+Cos[4 t]))^2))))^2)+(((((24 a ((Cos[t])^2) ((Sin[t])^2))/(3+Cos[4 t]))-((8 a ((Sin[t])^4))/(3+Cos[4 t]))+((32 a Cos[t] ((Sin[t])^3) Sin[4 t])/(((3+Cos[4 t]))^2))))^2)),a>0&&0<t<π]

4 a Integrate[(1/(((3+Cos[4 t]))^2))√(364+348 Cos[2 t]+304 Cos[4 t]+38 Cos[6 t]-28 Cos[8 t]-2 Cos[10 t]) Sin[t],{t,0,π/2}]

...blows memory...

Area

Parametric

Assuming[a>0,Area[curve,{t,0,π}]]

Inequality

Area[x^4+y^4-2a x y^2<0,{x,y}]

<<Graphics`InequalityGraphics`

InequalityPlot[x^4+y^4<2 x y^2,{x},{y},Ticks->None]

Integrate[Boole[x^4+y^4<2 x y^2],{x,-∞,∞},{y,-∞,∞}]//Timing

Integrate[Boole[x^4+y^4<2 x y^2],{y,-∞,∞},{x,-∞,∞}]

2Integrate[Boole[x^4+y^4<2 x y^2&&y>0],{x,-∞,∞},{y,-∞,∞}]

2Integrate[Boole[x^4+y^4<2 x y^2&&y>0],{y,-∞,∞},{x,-∞,∞}]

2NIntegrate[Boole[x^4+y^4<2 x y^2&&y>0],{x,-∞,∞},{y,-∞,∞}]

(1/2)NIntegrate[(((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])))^2,{θ,0,π}]

NIntegrate[(((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])))^2,{θ,0,π/2}]

(1/2)Integrate[(((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])))^2,{θ,0,π}]

Integrate[(((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])))^2,{θ,0,π/2}]

N[Pi/2/Sqrt[2]]

AreaInertiaTensor

Parametric

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

Polar

AreaInertiaTensor[((8a Cos[t] ((Sin[t])^2))/(3+Cos[4 t])),{t,0,π}]//Timing

Centroid

Parametric

Assuming[a>0,Centroid[curve,{t,0,π}]]

Numerical

NIntegrate[{x,y}Boole[(x^4)-2 x (y^2)+(y^4)<0],{x,0,1},{y,-1.5,1.5}]

Polar

Centroid[((8a Cos[t] ((Sin[t])^2))/(3+Cos[4 t])),{t,0,π}]//Timing

Inequality

eqn

eqn/.a->1

RegionPlot[(x^4)-2 x (y^2)+(y^4)<0,{x,-0,1},{y,-1.5,1.5},AspectRatio->Automatic,ImageSize->100,MaxRecursion->4]

V6

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]//FullSimplify//Timing

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0,GenerateConditions->True]//FullSimplify//Timing

N[(π/(4 (Sqrt[2])))]

V7

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]//FullSimplify//Timing

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0,GenerateConditions->True]//FullSimplify//Timing

(({(((a^3) π)/(4 (Sqrt[2]))),0})/(a^2 Pi/(2 Sqrt[2])))

V8

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]//FullSimplify//Timing

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0,GenerateConditions->True]//FullSimplify//Timing

Integrate[{x,y}Boole[(x^4)-2 a x (y^2)+(y^4)<0],{y,-∞,∞},{x,-∞,∞},Assumptions->a>0,GenerateConditions->True]//FullSimplify//Timing

Integrate[{x,y}Boole[(x^4)-2 x (y^2)+(y^4)<0],{y,-∞,∞},{x,-∞,∞},GenerateConditions->True]//FullSimplify//Timing

Integrate[{x,y}Boole[eqn<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]//FullSimplify//Timing

N[Last[%]]

(xybar=Integrate[{x,y}Boole[y^2(a^2-x^2)-(x^2+2a y-a^2)^2>0],{x,-a,a},{y,0,a},Assumptions->a>0]//FullSimplify)//Timing

(xybar=Integrate[{x,y}Boole[y^2(a^2-x^2)-(x^2+2a y-a^2)^2>0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]//FullSimplify)//Timing

(area=Integrate[Boole[y^2(a^2-x^2)-(x^2+2a y-a^2)^2>0],{x,-a,a},{y,0,a},Assumptions->a>0]//FullSimplify)//Timing

FullSimplify[(xybar/area)]

Curvature

V5?

FullSimplify[Curvature[((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])),θ],0<θ<π/2]

Plot[((8 (Sqrt[2]) (((3+Cos[4 θ]))^3) (3+2 Cos[2 θ]+Cos[4 θ]) Csc[θ])/(((182+174 Cos[2 θ]+152 Cos[4 θ]+19 Cos[6 θ]-14 Cos[8 θ]-Cos[10 θ]))^(3/2))),{θ,0,π},PlotStyle->Red]

V6

FullSimplify[Curvature[curve,t],a>0&&0<t<π]

SortBy[FullSimplify[Curvature[eqn,{x,y}],(x|y)∈Reals&&a>0],ByteCount]

ByteCount/@%

Polar

Simplify[Curvature[a(8  Cos[q] Sin[q]^2)/(3+Cos[4 q]),q],a>0&&0<q<π]

FullSimplify[%]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

FullSimplify[Slope[(8 a Cos[t] Sin[t]^2)/(3+Cos[4 t]) {Cos[t],Sin[t]},t]]

FullSimplify[Slope[x^4+y^4==2 a x y^2,{x,y}]]

FullSimplify[Slope[x^4+y^4==2 a x y^2,{x,y}]]

SpecialAffineCurvature

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

Tangential Angle

V5?

FullSimplify[TangentialAngle[((8 Cos[θ] ((Sin[θ])^2))/(3+Cos[4 θ])),θ],0<θ<π/2]

Plot[ArcTan[((4 Cos[θ] (3+Cos[4 θ]) Sin[θ])/(6+25 Cos[2 θ]+2 Cos[4 θ]-Cos[6 θ]))],{θ,0,π},PlotStyle->Red]

V6

FullSimplify[TangentialAngle[curve,t],0<t<π]

Plot[(1/2) (π-2 ArcTan[((4 (Cos[t]+Cos[3 t]) Csc[t])/(3+8 Cos[2 t]+Cos[4 t]))]),{t,0,π}]

Vector Properties

parm=(8 a Cos[t] Sin[t]^2)/(3+Cos[4 t]) {Cos[t],Sin[t]};
assum=a>0&&t∈Reals;
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