Dumbbell Curve

Author

Eric W. Weisstein
February 17, 2012

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

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

©2012 Wolfram Research, Inc. except for portions noted otherwise

Equations

Numerator[Together[Expand[y^2-((((a^2)-(x^2))(x^4))/(a^4))]]]

Factor[%]

eqn=-(a^2) (x^4)+(x^6)+(a^4) (y^2);

AlgebraicDegree[eqn,{x,y}]

y^2==a^2(x^4-x^6)/.{x->t,y->a t^2(Sqrt[1-t^2])}//Simplify

Eliminate[{x==t,(y^2)==(a^2)(t^4)(1-(t^2))},t]//FullSimplify

ParametricPlot[{t ,t^2Sqrt[1-t^2]},{t,-1,1}]

ParametricPlot[{t ,t^2Sqrt[1-t^2]},{t,0,1}]

Parametric

Refine[FullSimplify[Sqrt[((x^4)/(a^4))((a^2)-(x^2))]/.x->a t,-a<t<a&&a>0],a>0]

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[y^2==(x^4-x^6),{x,-5,5},AspectRatio->Automatic,Ticks->None,PlotStyle->Red]

Show[
Block[{$DisplayFunction=Identity},
Table[ImplicitPlot[y^2==a(x^4-x^6),{x,-5,5},AspectRatio->Automatic,Ticks->None,PlotStyle->Hue[a]],
{a,.05,1,.05}]]]

V6

ContourPlot[eqn[#][x,y]==0,{x,-1.,1.},{y,-.4,.4},AspectRatio->Automatic,ContourStyle->Red]&/@{.1,.5,1}

Properties

dumbbell={a t,a t^2Sqrt[1-t^2]};

Tangents

Maximize[t^2Sqrt[1-t^2],t]

Arc Length x(y)

ysoln=x/.Simplify[Solve[y^2==(x^4-x^6),x]]

ParametricPlot[Evaluate[{#,y}&/@ysoln],{y,-(2/(3(Sqrt[3]))),(2/(3(Sqrt[3])))},AspectRatio->Automatic,PlotStyle->List/@{Red,Orange,Yellow,Green,Blue,Purple}]

dy=(Sqrt[1+((D[#,y])^2)])&/@ysoln[[{2,4}]];

4Plus@@(NIntegrate[#,{y,0,(2/(3(Sqrt[3])))}]&/@dy)

dy2=Simplify[dy]

4Plus@@(NIntegrate[#,{y,0,(2/(3(Sqrt[3])))}]&/@dy2)

v4.2

4Plus@@(Integrate[#,{y,0,(2/(3(Sqrt[3])))}]&/@dy2)//Timing

...hang...

v5.1

4Plus@@(Integrate[#,{y,0,(2/(3(Sqrt[3])))}]&/@dy)//Timing

v6

4Plus@@(Integrate[#,{y,0,(2/(3(Sqrt[3])))}]&/@dy2)//Timing

Arc Length y(x)

Simplify[(Sqrt[1+((D[Sqrt[x^4-x^6],x])^2)])]

4NIntegrate[(Sqrt[((1+3 (x^2)-12 (x^4)+9 (x^6))/(1-(x^2)))]),{x,0,1}]

v4.2

4Integrate[(Sqrt[((1+3 (x^2)-12 (x^4)+9 (x^6))/(1-(x^2)))]),{x,0,1}]//Timing

v5.1

4Integrate[(Sqrt[((1+3 (x^2)-12 (x^4)+9 (x^6))/(1-(x^2)))]),{x,0,1}]//Timing

v6

4Integrate[(Sqrt[((1+3 (x^2)-12 (x^4)+9 (x^6))/(1-(x^2)))]),{x,0,1}]//Timing

Arc Length Parametrically

FullSimplify[(Sqrt[1+(((-((t^3)/(Sqrt[1-(t^2)]))+2 t (Sqrt[1-(t^2)])))^2)]),-1<t<1]

Numeric

2NIntegrate[(Sqrt[((1+3 (t^2)-12 (t^4)+9 (t^6))/(1-(t^2)))]),{t,-1,1}]

4NIntegrate[(Sqrt[((1+3 (t^2)-12 (t^4)+9 (t^6))/(1-(t^2)))]),{t,0,1}]

v5

ArcLength[dumbbell,t]

Integrate[(Sqrt[((1+3 (t^2)-12 (t^4)+9 (t^6))/(1-(t^2)))]),{t,0,1}]//Timing

v5.1

ArcLength[dumbbell,t]//Timing

v6

ArcLength[dumbbell,t]//Timing

2Integrate[(Sqrt[((1+3 (t^2)-12 (t^4)+9 (t^6))/(1-(t^2)))]),{t,-1,1}]//Timing

V6.1

Simplify[ArcLength[dumbbell,{t,0,1}],a>0]

FullSimplify[ArcLength[dumbbell/.a->1,{t,0,1}]]

4NIntegrate[(Sqrt[((1+3 (t^2)-12 (t^4)+9 (t^6))/(1-(t^2)))]),{t,0,1}]

Factor[1+3 (t^2)-12 (t^4)+9 (t^6)]

Area

V5

<<Graphics`InequalityGraphics`

InequalityPlot[y^2<x^4-x^6,{x},{y}]

V6

With[{a=1},RegionPlot[(a^4)y^2<((a^2)-(x^2))(x^4),{x,-1,1},{y,-.5,.5},MaxRecursion->5,AspectRatio->Automatic]]

Assuming[a>0,Area[(a^4)y^2<((a^2)-(x^2))(x^4),{x,y}]]//Timing

Assuming[a>0,Area[(a^4)y^2<((a^2)-(x^2))(x^4),{y,x}]]//Timing

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

Factor[(((x^4)/(a^2))-((x^6)/(a^4)))]

((((a^2)-(x^2))(x^4))/(a^4))

Integrate[Boole[y^2<a x^4-x^6],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]

Integrate[2(a^2) (Sqrt[(x^4)-(x^6)]),{x,-1,1}]

ParametricPlot[{{t,t^2Sqrt[1-t^2]},{t,-t^2Sqrt[1-t^2]}},{t,-1,1},AspectRatio->Automatic]

2Area[{t,t^2Sqrt[1-t^2]},{t,-1,1}]

AreaInertiaTensor

Assuming[a>0,AreaInertiaTensor[(a^4)y^2<((a^2)-(x^2))(x^4),{x,y}]]//Timing

Centroid

Assuming[a>0,Centroid[(a^4)y^2<((a^2)-(x^2))(x^4),{x,y}]]//Timing

Curvature

FullSimplify[Curvature[dumbbell,t],-1<t<1&&a>0]

Implicit Curvature

FullSimplify[Curvature[-(a^2) (x^4)+(x^6)+(a^4) (y^2),{x,y}],a>0]

FullSimplify[%,a>0]

Inflection Points

Simplify[{x,y}/.Solve[(a^2) (x^2) ((a^2) (x^4) (((2-3 (x^2)))^2)+3 (-2+5 (x^2)) (y^2))==eqn==0,{x,y}],a>0]//Union

pts={{-(1/2) (Sqrt[3-(Sqrt[(11/3)])]),-(1/12) (Sqrt[-21+5 (Sqrt[33])]) a},{-(1/2) (Sqrt[3-(Sqrt[(11/3)])]),(1/12) (Sqrt[-21+5 (Sqrt[33])]) a},{(1/2) (Sqrt[3-(Sqrt[(11/3)])]),-(1/12) (Sqrt[-21+5 (Sqrt[33])]) a},{(1/2) (Sqrt[3-(Sqrt[(11/3)])]),(1/12) (Sqrt[-21+5 (Sqrt[33])]) a}};

Block[{a=1},ContourPlot[eqn==0,{x,-2,2},{y,-2,2},Epilog->{Point[pts]}]]

InflectionPoints[eqn,{x,y}]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[y^2==(a^2-x^2)x^4/a^4,{x,y}]//FullSimplify

Tangential Angle

Simplify[((-((4 (t^2))/(Sqrt[1-(t^2)]))+2 (Sqrt[1-(t^2)])+(t^2) (-((t^2)/(((1-(t^2)))^(3/2)))-(1/(Sqrt[1-(t^2)]))))/(1+(((-((t^3)/(Sqrt[1-(t^2)]))+2 t (Sqrt[1-(t^2)])))^2)))]

V5.0

TangentialAngle[dumbbell,t]//Timing

V5.1

TangentialAngle[dumbbell,t]//Timing

Integrate[((2-9 (t^2)+6 (t^4))/((Sqrt[1-(t^2)]) (1+3 (t^2)-12 (t^4)+9 (t^6)))),{t,0,t}]//Timing

V6.0

V6.1

TangentialAngle[dumbbell,t]//Timing