Singular Point

Author

Eric W. Weisstein
March 16, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<Graphics`ImplicitPlot`

Cusp

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},{
{ImplicitPlot[x^3-y^2==0,{x,-.5,2},{y,-2,2},
Epilog->{PointSize[.1],Point[{0,0}]},
PlotStyle->Red,PlotLabel->StyleForm["cusp",FontFamily->"Times",FontSlant->"Italic"],Ticks->None,AspectRatio->Automatic,AxesOrigin->{0,0}],
ImplicitPlot[x^3-x^2+y^2==0,{x,-1,1.5},AxesOrigin->{0,0},Epilog->{PointSize[.1],Point[{0,0}]},Ticks->None,AspectRatio->Automatic,
PlotStyle->Red,PlotLabel->StyleForm["ordinary double point",FontFamily->"Times",FontSlant->"Italic"]],
ImplicitPlot[(x^2+y^2)^3-4x^2y^2==0,{x,-1,1},
PlotStyle->Red,PlotLabel->StyleForm["quadruple point",FontFamily->"Times",FontSlant->"Italic"],
Epilog->{PointSize[.1],Point[{0,0}]},Ticks->None,AspectRatio->Automatic,AxesOrigin->{0,0}]
},
{
ImplicitPlot[x^4+x^2y^2-2x^2y-x y^2+y^2==0,{x,-.1,1},PlotStyle->Red,PlotLabel->StyleForm["ramphoid cusp",FontFamily->"Times",FontSlant->"Italic"],
PlotRange->All,
Epilog->{PointSize[.1],Point[{0,0}]},Ticks->None,AspectRatio->Automatic,AxesOrigin->{0,0}],
ImplicitPlot[2x^4-3x^2y+y^2-2y^3+y^4==0,{x,-1.7,1.7},
PlotStyle->Red,PlotLabel->StyleForm["tacnode",FontFamily->"Times",FontSlant->"Italic"],Epilog->{PointSize[.1],Point[{0,0}]},Ticks->None,AspectRatio->Automatic,AxesOrigin->{0,0},PlotRange->All],
ImplicitPlot[(x^2+y^2)^2+3x^2 y-y^3==0,{x,-1,1},Ticks->None,
PlotStyle->Red,PlotLabel->StyleForm["triple point",FontFamily->"Times",FontSlant->"Italic"],AspectRatio->Automatic,AxesOrigin->{0,0},Epilog->{PointSize[.1],Point[{0,0}]}]
}
}]]]