Atriphtaloid

Author

Eric W. Weisstein
February 16, 2012

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

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

©2011 Wolfram Research, Inc. except for portions noted otherwise

Other Authors

Original version by Margherita Barile
June 18, 2002

Degree

eqn[a_,b_][x_,y_]:=x^4(x^2+y^2)-(a x^2-b)^2

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

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

AlgebraicDegree[eqn[a,b][x,y],{x,y}]

Plot

V5

<<Graphics`ImplicitPlot`

ImplicitPlot[x^4(x^2+y^2)-(2x^2-1)^2==0, {x,-2,2},PlotStyle->Red,Ticks->None]

Show[
Block[{$DisplayFunction=Identity},
Table[ImplicitPlot[x^4(x^2+y^2)-(a x^2-1)^2==0,{x,-5,5},AspectRatio->Automatic,Ticks->None,PlotStyle->Hue[a]],
{a,.2,3,.2}]],PlotRange->{{-3,3},{-3,3}}]

V6

ContourPlot[eqn[##][x,y]==0,{x,-3,3},{y,-3,3},AspectRatio->Automatic,ContourStyle->Red]&@@@{{1,1},{2,1},{3,1}}

Properties

Curvatures

ImplicitCurvature[eqn[a,b][x,y],{x,y}]

FullSimplify[%]

expr=SortBy[FullSimplify[Curvature[eqn[a,b][x,y],{x,y}],a>0&&b>0],ByteCount]

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]

κ=Factor[(x^2) ((x^4) (((2 a b-2 (a^2) (x^2)+3 (x^4)))^2)+(x^6) (-6 a b+2 (a^2) (x^2)+3 (x^4)) (y^2)-6 (x^8) (y^4))]

pts=Simplify[{x,y}/.Solve[κ==eqn==0,{x,y}],a>0&&b>0]

Length[pts]

{κ,eqn}/.Thread[{x,y}->pts[[1]]]//Simplify

{κ,eqn}/.Thread[{x,y}->pts[[2]]]//Simplify

N[pts/.{a->1,b->1}]

Singular Points

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

Reduce[eqn==D[eqn,x]==D[eqn,y]==0,{x,y},Reals]

Reduce[eqn==D[eqn,x]==D[eqn,y]==0/.{a->1,b->1},{x,y},Reals]

Slope

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

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