Superegg

Author

Eric W. Weisstein
May 9, 2008

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

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

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

Code

Superegg[{a_,b_,n_},{x_,y_,z_}]:=((Abs[(Sqrt[(((x^2)+(y^2))/(a^2))])])^n)+((Abs[(z/b)])^n)

Graphic

V5

Show[Block[{$DisplayFunction=Identity},
GraphicsArray[ContourPlot3D[Superegg[{1,2,#},{x,y,z}]-1,{x,-2,2},{y,-2,2},{z,-2,2},PlotLabel->TraditionalForm[n==#],PlotPoints->5,Mesh->None,Boxed->False,Axes->False]&/@{.5,1,1.5,2,3,4},ImageSize->400,GraphicsSpacing->-.25]
]]

V6

GraphicsGrid[{RegionPlot3D[Superegg[{1,2,#},{x,y,z}]<1,{x,-2,2},{y,-2,2},{z,-2,2},PlotLabel->n==#,Mesh->None,Boxed->False,Axes->False,PlotPoints->40]&/@{.5,1,1.5,2,3,4}},ImageSize->400,Spacings->-200]

GraphicsGrid[{ContourPlot3D[Superegg[{1,2,#},{x,y,z}]==1,{x,-2,2},{y,-2,2},{z,-2,2},PlotLabel->n==#,Mesh->None,Axes->False,Boxed->False]&/@{.5,1,1.5,2,3,4}},ImageSize->400,Spacings->-200]

Volume

Integrate[Boole[Superegg[{a,b,n},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0&&n>0]

n==1/2

FullSimplify[Superegg[{a,b,1/2},{x,y,z}],(x|y|z)∈Reals&&a>0&&b>0]

Integrate[Boole[Superegg[{a,b,(1/2)},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

Uses > 8 GB RAM

n==1

FullSimplify[Superegg[{a,b,1},{x,y,z}],(x|y|z)∈Reals&&a>0&&b>0]

Integrate[Boole[Superegg[{a,b,1},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

n==3/2

Integrate[Boole[Superegg[{a,b,3/2},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

n==2

FullSimplify[Superegg[{a,b,2},{x,y,z}],(x|y|z)∈Reals&&a>0&&b>0]

Integrate[Boole[Superegg[{a,b,2},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

n==3

Integrate[Boole[Superegg[{a,b,3},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

n==4

FullSimplify[Superegg[{a,b,4},{x,y,z}],(x|y|z)∈Reals&&a>0&&b>0]

Integrate[Boole[Superegg[{a,b,4},{x,y,z}]<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&b>0]//Timing

((2 b)/a) Integrate[((((a^4)-((((x^2)+(y^2)))^2)))^(1/4)),{x,-a,a},{y,-(Sqrt[(a^2)-(x^2)]),(Sqrt[(a^2)-(x^2)])},Assumptions->a>0]//Timing