Pear-Shaped Curve

Author

Eric W. Weisstein
February 20, 2012

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/Pear-ShapedCurve.nb.

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

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

Definition

eqn[a_,b_][x_,y_]:=(b^2)(y^2)-(x^3)(a-x)

Equation

Solve[(b^2)(y^2)==(x^3)(a-x),y]

Expand[eqn[a,b][x,y]]

eqn=-a (x^3)+(x^4)+(b^2) (y^2);

AlgebraicDegree[%,{x,y}]

Plot

V5

Cartesian

With[{a=1,b=1},
Plot[Evaluate[(((Sqrt[a-x]) (x^(3/2)))/b){-1,1}],{x,0,1},Ticks->None,PlotStyle->Red,AspectRatio->Automatic]
]

Implicit

<<Graphics`ImplicitPlot`

With[{a=1,b=1},
ImplicitPlot[(b^2)(y^2)==(x^3)(a-x),{x,-2,2},PlotStyle->Red,Ticks->None]
]

V6

ContourPlot[eqn[##][x,y]==0,{x,-.1,2.1},{y,-1.4,1.4},AspectRatio->Automatic,ContourStyle->Red]&@@@{{2,1},{1,1},{1,2}}

Properties

ArcLength

V5

ArcLength[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x]//Timing

2ArcLength[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},{x,0,1}]//Timing

V6

ArcLength[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x]//Timing

2ArcLength[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},{x,0,1}]//Timing

Area

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

2Simplify[Abs[Area[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},{x,0,a}]],a>0&&b>0]

<<Graphics`InequalityGraphics`

With[{a=1,b=1},
InequalityPlot[(b^2)(y^2)<(x^3)(a-x),{x},{y}]
]

<<Calculus`Integration`

With[{a=1,b=1},Integrate[Boole[(b^2)(y^2)<(x^3)(a-x)],{x},{y}]
]

AreaInertiaTensor

Assuming[a>0&&b>0,AreaInertiaTensor[-a (x^3)+(x^4)+(b^2) (y^2)<0,{x,y}]]//Timing

Centroid

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

Curvature

v5

FullSimplify[Curvature[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x],b>0&&a>0&&0<x<1]

v6

FullSimplify[Curvature[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x],b>0&&a>0&&0<x<1]

Implicit Curvature

FullSimplify[ImplicitCurvature[(b^2)(y^2)-(x^3)(a-x),{x,y}]]

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

Inflection Points

Simplify[{x,y}/.Solve[eqn==2 (b^2) (((3 a-4 x))^2) (x^4)-24 (b^4) (a-2 x) x (y^2)==0,{x,y}],a>0&&b>0]//Union

InflectionPoints[eqn,{x,y}]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

Slope[a^2==x^2+2 x^3+5 x^4+6 x^5+6 x^6+4 x^7+x^8+y^2+2 x y^2+2 x^2 y^2+4 x^3 y^2+14 x^4 y^2+12 x^5 y^2+4 x^6 y^2-3 y^4-2 x y^4+10 x^2 y^4+12 x^3 y^4+6 x^4 y^4+2 y^6+4 x y^6+4 x^2 y^6+y^8,{x,y}]//FullSimplify

Tangential Angle

V5

TangentialAngle[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x]//Timing

V6

TangentialAngle[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x]//Timing

...crashes (because of arc length crash)...

V9

TangentialAngle[{x,(((Sqrt[a-x]) (x^(3/2)))/b)},x]//Timing