Cassini Ovals

Author

Eric W. Weisstein
February 17, 2012

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

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

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

Equation

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

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

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

AlgebraicDegree[eqn,{x,y}]

Expand[eqn-((x^2+y^2+a^2)^2-b^4-4a^2x^2)]

Plot

V5

<<Graphics`ImplicitPlot`

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},ImplicitPlot[(x^2+y^2+a^2)^2-4a^2x^2==b^4/.Thread[{a,b}->#],{x,-5,5},PlotStyle->Red,PlotPoints->50,Ticks->None]&/@({1,#}&/@{1.5,1.1,1,.95})]
]]

Show[
Block[{$DisplayFunction=Identity},Table[
ImplicitPlot[(x^2+y^2+a^2)^2-4a^2x^2==b^4/.a->1,{x,-5,5},PlotStyle->Hue[b/1.5],PlotPoints->50,Ticks->None],{b,.1,1.5,.1}]
]
]

V6

parms=({1,#}&/@{.8,.95,1,1.05,1.2,1.5})

ContourPlot[eqn[##][x,y]==0,{x,-2,2},{y,-1.2,1.2},AspectRatio->Automatic,ContourStyle->Red]&@@@parms

parms=({1.2,#}&/@{.8,.95,1,1.05,1.2,1.5})

ContourPlot[eqn[##][x,y]==0,{x,-4,4},{y,-1.2,1.2},AspectRatio->Automatic,ContourStyle->Red]&@@@parms

Function[{a,b},Piecewise[{{{{0,0}->"Crunode"},a==b}},None]][1,1]

Properties

Area

V6?

Integrate[Boole[(a^4)-(b^4)-2 (a^2) (x^2)+(x^4)+2 (a^2) (y^2)+2 (x^2) (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&b>0]//Timing

V8

Integrate[Boole[(a^4)-(b^4)-2 (a^2) (x^2)+(x^4)+2 (a^2) (y^2)+2 (x^2) (y^2)+(y^4)<0],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0&&b>0]//Timing

Integrate[Boole[eqn[a,b][x,y]<0],{x,-∞,∞},{y,-∞,∞},Assumptions->0<b<a]//Timing

Integrate[Boole[eqn[a,b][x,y]<0],{x,-∞,∞},{y,-∞,∞},Assumptions->0<a<b]//Timing

Explicit

a^2 Integrate[Cos[2t]+Sqrt[c^4-Sin[2t]^2]/. c->(b/a),{t,-Pi/4,Pi/4},Assumptions->a>0&&b>0] //Timing

a^2 Integrate[Cos[2t]+Sqrt[c^4-Sin[2t]^2]/. c->(b/a),{t,-Pi/4,Pi/4},Assumptions->0<b<a] //Timing

a^2 Integrate[Cos[2t]+Sqrt[c^4-Sin[2t]^2]/. c->(b/a),{t,-Pi/4,Pi/4},Assumptions->0<a<b] //Simplify//Timing

a = b

Limit[(a^2)+(b^2) EllipticE[((a^4)/(b^4))],b->a]

4(1/2)2a^2 Integrate[Cos[2θ],{θ,0,π/4}]

<<Graphics`Graphics`
<<Graphics`Colors`

PolarPlot[Sqrt[2Cos[2θ]],{θ,-Pi/4,Pi/4},Ticks->None,PlotStyle->Red]

Curvature

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

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

ByteCount/@%

Slope

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

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

Region

RegionPlot[eqn[##][x,y]<0,{x,-2,2},{y,-1.2,1.2}]&@@@parms