Stirrup Curve

Author

Eric W. Weisstein
February 20, 2012

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

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

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

Definition

eqn=(x^2-1)^2-y^2(y-1)(y-2)(y+5);

Equation

Expand[eqn]

AlgebraicDegree[eqn,{x,y}]

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[(x^2-1)^2==y^2(y-1)(y-2)(y+5),{x,-4,4},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

V6

ContourPlot[eqn,{x,-5,5},{y,-5,5},ContourStyle->Red,AspectRatio->Automatic]

Properties

Area

RegionPlot[(x^2-1)^2<y^2(y-1)(y-2)(y+5),{x,-5,5},{y,-5,5}]

RegionPlot[(x^2-1)^2<y^2(y-1)(y-2)(y+5),{x,-2,2},{y,-2,2}]

RegionPlot[(x^2-1)^2<y^2(y-1)(y-2)(y+5)&&y>0,{x,-2,2},{y,-.4,1.5},MaxRecursion->3]

Integrate[Boole[(x^2-1)^2<y^2(y-1)(y-2)(y+5)&&y>0],{x,-∞,∞},{y,-4/10,3/2}]

RegionPlot[(x^2-1)^2>y^2(y-1)(y-2)(y+5)&&-1<x<1&&-1<y<6/10,{x,-1,1},{y,-.4,.5}]

Integrate[Boole[(x^2-1)^2>y^2(y-1)(y-2)(y+5)&&-1<x<1&&-1<y<6/10],{x,-∞,∞},{y,-∞,∞}]

NIntegrate[Boole[(x^2-1)^2>y^2(y-1)(y-2)(y+5)&&-1<x<1&&-1<y<6/10],{x,-∞,∞},{y,-∞,∞}]

Implicit Curvature

FullSimplify[ImplicitCurvature[eqn,{x,y}],(x|y)∈Reals]

Inflection Points

infl=InflectionPoints[eqn,{x,y}]//FullSimplify

Singular Points

sing=SingularPoints[eqn,{x,y}]

ContourPlot[eqn==0,{x,-5,5},{y,-5,5},Epilog->{PointSize[.02],{Red,Point[infl]},Point[sing]}]

ContourPlot[eqn==0,{x,-8,8},{y,0,4},Epilog->{PointSize[.02],{Red,Point[infl]},Point[sing]}]

ContourPlot[eqn==0,{x,-1,1},{y,-5,-4.999},Epilog->{PointSize[.02],{Red,Point[infl]},Point[sing]}]

Slope

Slope[(x^2-1)^2==y^2 (y-1) (y-2) (y+5),{x,y}]//FullSimplify