Knot Curve

Author

Eric W. Weisstein
February 2, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Equation

eqn=(x^2-(a^2))^2==a y^2(3a+2y);

Expand[Subtract@@eqn]

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[(x^2-1)^2==y^2(3+2y),{x,-2,2},PlotPoints->10,PlotStyle->Red,Ticks->None,AspectRatio->Automatic]

V6

ContourPlot[(x^2-1)^2==y^2(3+2y),{x,-2,2},{y,-1.6,1.3},AspectRatio->Automatic]

RegionPlot[(x^2-1)^2<y^2(3+2y),{x,-2,2},{y,-1.6,1.3},AspectRatio->Automatic]

Properties

reg={RegionPlot[(x^2-1)^2<y^2(3+2y)&&y<0,{x,-2,2},{y,-1.6,1.3},AspectRatio->Automatic],RegionPlot[(x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1,{x,-2,2},{y,-1.6,1.3},AspectRatio->Automatic]}

RegionPlot[((x^2-1)^2<y^2(3+2y)&&y<0)||((x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1),{x,-2,2},{y,-1.6,1.3},AspectRatio->Automatic]

Area

NIntegrate[Boole[((x^2-1)^2<y^2(3+2y)&&y<0)||((x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1)],{x,-∞,∞},{y,-∞,∞}]

Left lobe

Integrate[Boole[(x^2-1)^2<y^2(3+2y)&&y<0],{x,0,∞},{y,-∞,∞}]

area1=Integrate[Boole[(x^2-1)^2<y^2(3+2y)&&y<0],{y,-∞,∞},{x,0,∞}]

Half of central wegde

Integrate[Boole[(x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1],{x,0,∞},{y,-∞,∞}]

area2=Integrate[Boole[(x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1],{y,-∞,∞},{x,0,∞}]

Total

FullSimplify[2area1+2area2]

N[%]

Intercepts

Solve[eqn/.x->0,y]

Solve[eqn/.y->0,x]

ImplicitCurvature

SortBy[FullSimplify[Curvature[eqn,{x,y}],a>0],ByteCount]

ByteCount/@%

Singular Points

SingularPoints[Subtract@@eqn,{x,y}]

Slope

FullSimplify[Slope[eqn,{x,y}],a>0]//Union

Solve y(x)

Curves

xsoln=y/.Solve[(x^2-1)^2==y^2(3+2y),y]

Plot

Plot[Evaluate[soln],{x,-2,2},PlotStyle->List/@{Red,Green,Blue},AspectRatio->Automatic]

Solve[soln[[1]]==soln[[3]],x]

Simplify[soln[[1]]/.x->(Sqrt[2])]

Solve x(y)

ysoln=x/.Solve[(x^2-1)^2==y^2(3+2y),x]

Solve[-(Sqrt[1-y (Sqrt[3+2 y])])==-(Sqrt[1+y (Sqrt[3+2 y])]),y]

ParametricPlot[Evaluate[{#,y}&/@{-(Sqrt[1-y (Sqrt[3+2 y])]),(Sqrt[1-y (Sqrt[3+2 y])]),-(Sqrt[1+y (Sqrt[3+2 y])]),(Sqrt[1+y (Sqrt[3+2 y])])}],{y,-2,1},PlotRange->All,PlotStyle->{Red,Yellow,Green,Blue}]

Tangents

Vertical

Last[Maximize[{x,(x^2-1)^2==y^2(3+2y)&&y<0},{x,y}]]

Horizontal

Last[Maximize[{y,(x^2-1)^2==y^2(3+2y)&&-1<x<1},{x,y}]]

Last[Minimize[{y,(x^2-1)^2==y^2(3+2y)&&y<0},{x,y}]]

Area y(x)

Region 1

f=FullSimplify[soln[[3]]-soln[[1]]]

NIntegrate[f,{x,0,(Sqrt[2])},MaxRecursion->20,WorkingPrecision->20]

Integrate[f,{x,0,(Sqrt[2])}]//Timing

Region 2

f=FullSimplify[soln[[2]]-soln[[3]],0<x<1]

2NIntegrate[f,{x,0,1}]

2Integrate[f,{x,0,1}]//Timing

Area x(y)

FullSimplify[-(Sqrt[1-y (Sqrt[3+2 y])])+(Sqrt[1+y (Sqrt[3+2 y])])]

Region 1

-Simplify[Integrate[(-(Sqrt[1-y (Sqrt[3+2 y])])+(Sqrt[1+y (Sqrt[3+2 y])])),{y,-3/2,0}]]

N[%]

Region 2

{-(Sqrt[1-y (Sqrt[3+2 y])]),(Sqrt[1-y (Sqrt[3+2 y])]),-(Sqrt[1+y (Sqrt[3+2 y])]),(Sqrt[1+y (Sqrt[3+2 y])])}

FullSimplify[-(Sqrt[1+y (Sqrt[3+2 y])])+(Sqrt[1-y (Sqrt[3+2 y])])]

FullSimplify[2(Integrate[(Sqrt[1+y (Sqrt[3+2 y])]),{y,-1,0}]+Integrate[(Sqrt[1-y (Sqrt[3+2 y])]),{y,0,1/2}])]

N[%]

Total

Simplify[2(12/7) (Sqrt[3]) (-1+(Sqrt[2]))-(12/7) (Sqrt[3]) (-2+(Sqrt[2]))]

Plot

<<Graphics`InequalityGraphics`
<<Graphics`ImplicitPlot`

Show[
Block[{$DisplayFunction=Identity},
{
InequalityPlot[(x^2-1)^2>y^2(3+2y)&&-1<x<1&&y>-1,{x,-2,2},{y,-10,10},Curves->None,Fills->Orange],
InequalityPlot[(x^2-1)^2<y^2(3+2y)&&x<0&&y<0,{x,-2,2},{y,-10,10},Curves->None,Fills->Yellow],
ImplicitPlot[(x^2-1)^2==y^2(3+2y),{x,-2,2},PlotPoints->10,PlotStyle->{{Red}}],
Graphics[{
Text["(Subscript[A, 2])",{0,-.2},Background->White,TextStyle->{FontFamily->"Times",FontSlant->"Italic"}],
Text["(Subscript[A, 1])",{-.9,-.9},TextStyle->{FontFamily->"Times",FontSlant->"Italic"}]
}]
}],
Ticks->None,AspectRatio->Automatic,PlotRange->{{-1.5,1.5},{-1.5,.6}}
]