Tooth Surface

Author

Eric W. Weisstein
June 18, 2008

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

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

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

Initialization

Tooth:=x^4+y^4+z^4-(x^2+y^2+z^2)

Plot

MaxValue[{x,x^4+y^4+z^4-(x^2+y^2+z^2)==0},{x,y,z}]//FullSimplify

N[%]

With[{r=1.2},ContourPlot3D[Tooth==0,{x,-r,r},{y,-r,r},{z,-r,r},Boxed->False,Axes->False,PlotPoints->20]]

With[{r=1.2},ContourPlot3D[Tooth==0,{x,-r,r},{y,-r,r},{z,-r,r},PlotPoints->20]]

With[{r=1.2},RegionPlot3D[Tooth<0,{x,-r,r},{y,-r,r},{z,-r,r}]]

Properties

Bounding Box

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

N[%]

r=-Root[-1-2 ((#1)^2)+2 ((#1)^4)&,1];

Gaussian Curvature

SortBy[FullSimplify[GaussianCurvature[Tooth,{x,y,z}],(x|y|z)∈Reals],ByteCount]

ByteCount/@%

Volume

Cartesian

With[{r=-Root[-1-2 ((#1)^2)+2 ((#1)^4)&,1]},NIntegrate[Boole[x^4+y^4+z^4-(x^2+y^2+z^2)<0],{x,-r,r},{y,-r,r},{z,-r,r}]]//Timing

8NIntegrate[Boole[x^4+y^4+z^4<(x^2+y^2+z^2)],{x,0,1.2},{y,0,1.2},{z,0,1.2},Method->"QuasiMonteCarlo"]

8NIntegrate[Boole[x^4+y^4+z^4<(x^2+y^2+z^2)],{x,0,1.2},{y,0,1.2},{z,0,1.2},Method->"QuasiMonteCarlo",PrecisionGoal->4,MaxPoints->10^6]//Timing

Integrate[Boole[Tooth<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]//Timing

i=Integrate[Boole[Tooth<0],{x,-r,r}]

FullSimplify[i,-r<=y<=r&&-r<=z<=r]

Cylindrical

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

NIntegrate[Boole[((3 (r^4))/4)+(z^4)+(1/4) (r^4) Cos[4 θ]<(r^2)+(z^2)]r,{r,0,1.2(Sqrt[2])},{z,-1.2,1.2},{θ,0,2π}]//Timing

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

NIntegrate[Boole[-(r^2)+((3 (r^4))/4)-(z^2)+(z^4)+(1/4) (r^4) Cos[4 θ]<0]r,{r,0,1.2(Sqrt[2])},{z,-1.2,1.2},{θ,0,2π}]//Timing

Integrate[Boole[-(r^2)+((3 (r^4))/4)-(z^2)+(z^4)+(1/4) (r^4) Cos[4 θ]<0]r,{r,0,∞},{z,-∞,∞},{θ,0,2π}]//Timing

Integrate[Boole[-(r^2)+((3 (r^4))/4)-(z^2)+(z^4)+(1/4) (r^4) Cos[4 θ]<0]r,{z,-∞,∞},{r,0,∞},{θ,0,2π}]//Timing

Spherical

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

NIntegrate[Boole[(-8 ((Cos[ϕ])^2)+4 (1+(Sqrt[3])) ((Cos[ϕ])^4)-8 ((Sin[ϕ])^2)+(1+(Sqrt[3])) (3+Cos[4 θ]) ((Sin[ϕ])^4))<0](r^2)Sin[ϕ],{r,0,1.2(Sqrt[3])},{ϕ,0,π},{θ,0,2π}]//Timing

NIntegrate[Boole[(-8 ((Cos[ϕ])^2)+4 (1+(Sqrt[3])) ((Cos[ϕ])^4)-8 ((Sin[ϕ])^2)+(1+(Sqrt[3])) (3+Cos[4 θ]) ((Sin[ϕ])^4))<0](r^2)Sin[ϕ],{r,0,1.2(Sqrt[3])},{ϕ,0,π},{θ,0,2π},Method->"AdaptiveQuasiMonteCarlo"]//Timing

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

NIntegrate[Boole[(1/8) (1+(Sqrt[3])) (4 ((Cos[ϕ])^4)+(3+Cos[4 θ]) ((Sin[ϕ])^4))<1](r^2)Sin[ϕ],{r,0,1.2(Sqrt[3])},{ϕ,0,π},{θ,0,2π}]//Timing

POVRay

<<Utilities`POVray`

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