Hunt's Surface

Author

Eric W. Weisstein
June 10, 2008

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

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

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

Equation

eqn=4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2;

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

Plot

surf=Block[{r=4},ContourPlot3D[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2==0,{x,-r,r},{y,-r,r},{z,-r,r},Boxed->False,Mesh->False,Axes->False,MaxRecursion->3,PlotPoints->50]]

RegionPlot3D[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2<0,{x,-4,4},{y,-4,4},{z,-4,4},PlotPoints->50,MaxRecursion->4]

Properties

Bounding Box

Through[{MinValue,MaxValue}[{#,eqn==0},{x,y,z}]]&/@{x,y,z}

N[%]

Through[{MinValue,MaxValue}[{#,eqn==0},{x,y,z}]]&/@{(Sqrt[(x^2)+(y^2)])}

Gaussian Curvature

SortBy[FullSimplify[GaussianCurvature[eqn,{x,y,z}]],ByteCount]

ByteCount/@%

Singular Points

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

real=Select[pts,Im[#]==1.{0,0,0}&]//Union

Show[{surf,Graphics3D[{PointSize[.02],Point[real]}]}]

Volume

Cartesian

NIntegrate[Boole[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2<0],{x,-(8/(Sqrt[7])),(8/(Sqrt[7]))},{y,-(8/(Sqrt[5])),(8/(Sqrt[5]))},{z,-3 (Sqrt[(3/7)]),3 (Sqrt[(3/7)])}]//Timing

Integrate[Boole[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2<0],{x,-4,4},{y,-4,4},{z,-4,4}]//Timing

Integrate[Boole[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2<0],{z,-4,4},{x,-4,4},{y,-4,4}]//Timing

Cylindrical

FullSimplify[4(x^2+y^2+z^2-13)^3+27(3x^2+y^2-4z^2-12)^2/.{x->r Cos[θ],y->r Sin[θ]},r>0&&0<θ<2π]

NIntegrate[Boole[4 (((-13+(r^2)+(z^2)))^3)+27 (((2 ((r^2)-2 (3+(z^2)))+(r^2) Cos[2 θ]))^2)<0]r,{θ,0,2π},{r,0,(8/(Sqrt[5]))},{z,-3 (Sqrt[(3/7)]),3 (Sqrt[(3/7)])}]//Timing

Integrate[Boole[4 (((-13+(r^2)+(z^2)))^3)+27 (((2 ((r^2)-2 (3+(z^2)))+(r^2) Cos[2 θ]))^2)<0]r,{θ,0,2π},{r,0,(8/(Sqrt[5]))},{z,-3 (Sqrt[(3/7)]),3 (Sqrt[(3/7)])}]//Timing

Integrate[Boole[4 (((-13+(r^2)+(z^2)))^3)+27 (((2 ((r^2)-2 (3+(z^2)))+(r^2) Cos[2 θ]))^2)<0]r,{θ,0,2π},{z,-3 (Sqrt[(3/7)]),3 (Sqrt[(3/7)])},{r,0,(8/(Sqrt[5]))}]//Timing

POVRay

<<Utilities`POVRay`

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

my={4, 0, 0, 0, 12, 0, 0, 12, 0, 87, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 24, 0, -150, 0, 0, 0, 0, 12, 0, -960, 0, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 0, -129, 0, 0, 0, 0, 12, 0, -528, 0, 1380, 0, 0, 0, 0, 0, 0, 4, 0, 276, 0, 4620, 0, -4900};

nordstrand={4.,0,0,0,12.,0,0,12.,0,87.,0,0,0,0,0,0,0,0,0,0,12.,0,0,24.,0,-150.,0,0,0,0,12.,0,-960.,0,84.,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4.,0,0,12.,0,-129.,0,0,0,0,12.,0,-528.,0,1380.,0,0,0,0,0,0,4.,0,276.,0,4620.,0,-4900.000000000001};

my-nordstrand