Capricornoid

Author

Eric W. Weisstein
February 17, 2012

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

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

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

Additional Authors

Original notebook by Margherita Barile
June 21, 2002

Degree

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

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

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

AlgebraicDegree[eqn,{x,y}]

Plot

V5

<<Graphics`ImplicitPlot`

With[{a=2,b=8},
ImplicitPlot[a^2x^2(x^2+y^2)-b(a y-x^2-y^2)^2==0, {x,-3,3},AspectRatio->Automatic,PlotRange->All,
PlotStyle->Red,Ticks->None]
]

V6

ContourPlot[eqn[2,8][x,y]==0,{x,-2.4,2.4},{y,-.1,2.5},AspectRatio->Automatic,MaxRecursion->6]

ContourPlot[eqn[##][x,y]==0,{x,-2.4,2.4},{y,-1,3},AspectRatio->Automatic,MaxRecursion->5]&@@@{{1,2},{2,8},{2,1},{1,5},{1,8},{1,30}}

Block[{a=2,b=1},RegionPlot[a^2x^2(x^2+y^2)-b(a y-x^2-y^2)^2<0,{x,-1.2,1.2},{y,-1,3},AspectRatio->Automatic]]

Block[{a=1,b=2},RegionPlot[a^2x^2(x^2+y^2)-b(a y-x^2-y^2)^2>0,{x,-1.2,1.2},{y,0,1.2},AspectRatio->Automatic]]

Properties

Curvature

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

FullSimplify[%,a>0&&b>0]

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

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]

Singular Points

{x,y}/.Solve[eqn==((a^2) b (y^2) (-2 (b^2) y ((x^2)+(y^2))-2 (a^2) b y (b+(x^2)+(y^2))+(a^3) (2 b (x^2)+(b-(x^2)) (y^2))+a b (2 b (x^2)+2 (2 b+(x^2)) (y^2)+(y^4))))==0,{x,y}]

SingularPoints[eqn,{x,y}]

Slope

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

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