Cassini Surface

Author

Eric W. Weisstein
June 3, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/CassiniSurface.nb.

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

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

Definition

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

Equation

AlgebraicDegree[eqn,{x,y,z}]

Surface

V5

a=1.;
d=2.;
Show[{Plot3D[(((x-a)^2+y^2)((x+a)^2+y^2))^(1/4),{x,-d,d},{y,-d,d},PlotPoints->30,DisplayFunction->Identity],Plot3D[-(((x-a)^2+y^2)((x+a)^2+y^2))^(1/4),{x,-d,d},{y,-d,d},PlotPoints->30,DisplayFunction->Identity]},Boxed->False,Axes->False,DisplayFunction->$DisplayFunction,AspectRatio->1,ViewPoint->{3,3,2}]

a=1.;
d=2.;
Show[
  {
  Plot3D[(((x-a)^2+y^2)((x+a)^2+y^2))^(1/4),{x,-d,d},{y,-d,d},
    PlotPoints->40,DisplayFunction->Identity],
  Plot3D[-(((x-a)^2+y^2)((x+a)^2+y^2))^(1/4),{x,-d,d},{y,-d,d},
    PlotPoints->40,DisplayFunction->Identity]
  },
  Boxed->False,Axes->False,AspectRatio->1,DisplayFunction->$DisplayFunction,ViewPoint->{0,3,0}
]

cassini=Expand[((x-a)^2+y^2)((x+a)^2+y^2)-z^4]

FullSimplify[cassini /. y->0]

V6

With[{r=2},ContourPlot3D[((((x-1))^2)+(y^2))((((x+1))^2)+(y^2))==(z^4),{x,-r,r},{y,-r,r},{z,-r,r},RegionFunction->(#1^2+#2^2+#3^2<r^2&),Boxed->False,Axes->False]]

With[{r=2},ContourPlot3D[((((x-.5))^2)+(y^2))((((x+.5))^2)+(y^2))==(z^4),{x,-r,r},{y,-r,r},{z,-r,r},RegionFunction->(#1^2+#2^2+#3^2<r^2&),Boxed->False,Axes->False]]

With[{r=2},ContourPlot3D[((((x-.2))^2)+(y^2))((((x+.2))^2)+(y^2))==(z^4),{x,-r,r},{y,-r,r},{z,-r,r},RegionFunction->(#1^2+#2^2+#3^2<r^2&),Boxed->False,Axes->False]]

POVRay

<<Utilities`POVRay`

POVCoefs[cassini]

Properties

FullSimplify[GaussianCurvature[eqn,{x,y,z}],z>0]