Goursat's Surface

Author

Eric W. Weisstein
November 11, 2008

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

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

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

Definition

Goursat[a_,b_,c_][x_,y_,z_]:=x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c

Expand[Goursat[a,b,c][x,y,z]]

Graphics

V5

<<Utilities`LiveGraphics3D`

<<ImplicitPlot3D`

WriteLiveForm["gourcube.m",
ImplicitPlot3D[x^4+y^4+z^4==1,{x,-1,1},{y,-1,1},{z,-1,1},Boxed->False]]

WriteLiveForm["goursurf.m",
ImplicitPlot3D[
x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c /.{a->0,b->-2,c->-1},{x,-2,2},{y,-2,2},{z,-2,2},Boxed->False]]

V7

MakeBoxes[label[{a_,b_,c_}],TraditionalForm]:=RowBox[{RowBox[{RowBox[{"a","==",MakeBoxes[a,TraditionalForm]}],",",RowBox[{"b","==",MakeBoxes[b,TraditionalForm]}],",",RowBox[{"c","==",MakeBoxes[c,TraditionalForm]}]}]}]

GraphicsGrid[{ContourPlot3D[Goursat[Sequence@@#][x,y,z]==0,{x,-2,2},{y,-2,2},{z,-2,2},Boxed->False,PlotLabel->label[#],Axes->False]&/@{{1,0,-1},{0,0,-1},{0,-2,-1},{0,-1,1/2},{-1,1,-1}}},ImageSize->400]

GraphicsGrid[Partition[ContourPlot3D[Goursat[Sequence@@#][x,y,z]==0,{x,-2,2},{y,-2,2},{z,-2,2},Boxed->False,PlotLabel->label[#],Axes->False,Method->{"ShrinkWrap"->True}]&/@{{1,0,-1},{0,0,-1},{0,-2,-1},{0,-1,1/2},{-1,1,-1}},3,3,{1,1},{}],ImageSize->400]

GraphicsGrid[Partition[Labeled[ContourPlot3D[Goursat[Sequence@@#][x,y,z]==0,{x,-2,2},{y,-2,2},{z,-2,2},Boxed->False,Axes->False,PlotRange->All,Method->{"ShrinkWrap"->True}],TraditionalForm[label[#]],Top]&/@{{1,0,-1},{0,0,-1},{0,-2,-1},{0,-2,1},{0,-1,1/2},{-1,1,-1}},3,3,{1,1},{}],ImageSize->400,ItemAspectRatio->1.3]

POV-Ray

<<POVray`

POVCoefficients[x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c /.{a->0,b->0,c->-1},{x,y,z}]

POVCoefficients[x^4+y^4+z^4+a(x^2+y^2+z^2)^2+b(x^2+y^2+z^2)+c /. {a->0,b->-2,c->-1},{x,y,z}]

Properties

ByteCount/@(k=GaussianCurvature[Goursat[a,b,c][x,y,z],{x,y,z}])//Timing

FullSimplify[k[[1]],a>0&&b>0&&c>0]

Volume

Integrate[Boole[Goursat[a,b,c][x,y,z]<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->(a|b|c)∈Reals]//Timing

{a==1,b==0,c==-1}

surf=Goursat[1,0,-1][x,y,z]

Cartesian

Maximize[{x,surf==0},{x,y,z}]//FullSimplify

N[%]

RegionPlot3D[surf<0,{x,-1,1},{y,-1,1},{z,-1,1}]

NIntegrate[Boole[surf<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->"AdaptiveQuasiMonteCarlo"]

Integrate[Boole[surf<0],{#1,0,1},{#2,0,1},{#3,0,1}]&@@@Permutations[{x,y,z}]

ToRadicals[Root[-1+2 (x^4)+2 (x^2) (y^2)+2 (y^4)+(2 (x^2)+2 (y^2)) ((#1)^2)+2 ((#1)^4)&,2]]

Plot3D[Abs[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2]))-Root[-1+2 (x^4)+2 (x^2) (y^2)+2 (y^4)+(2 (x^2)+2 (y^2)) ((#1)^2)+2 ((#1)^4)&,2]],{x,0,(2^(-1/4))},{y,0,(2^(-1/4))}]

8NIntegrate[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2])), 
  {x, 0, Root[-1 + 2*#1^4 & , 2]}, {y, 0, Root[-1 + 2*x^4 + 2*x^2*#1^2 + 2*#1^4 & , 2]}]

ToRadicals[Root[-1 + 2*#1^4 & , 2]]

8NIntegrate[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2])), 
  {x, 0, (2^(-1/4))}, {y, 0, Root[-1 + 2*x^4 + 2*x^2*#1^2 + 2*#1^4 & , 2]}]

ToRadicals[ Root[-1 + 2*x^4 + 2*x^2*#1^2 + 2*#1^4 & , 2]]

8NIntegrate[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2])), {x, 0, (2^(-1/4))}, {y, 0, ((Sqrt[-(x^2)+(Sqrt[2-3 (x^4)])])/(Sqrt[2]))}]

8Integrate[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2])), {x, 0, (2^(-1/4))}, {y, 0, ((Sqrt[-(x^2)+(Sqrt[2-3 (x^4)])])/(Sqrt[2]))}]

8Integrate[((Sqrt[-(x^2)-(y^2)+(Sqrt[2-3 (x^4)-2 (x^2) (y^2)-3 (y^4)])])/(Sqrt[2])), {y, 0, ((Sqrt[-(x^2)+(Sqrt[2-3 (x^4)])])/(Sqrt[2]))},Assumptions->0<x<(2^(-1/4))]

Spherical

FullSimplify[-1+(x^4)+(y^4)+(z^4)+((((x^2)+(y^2)+(z^2)))^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]

Solve[-1+(r^4)+(r^4) (((Cos[ϕ])^4)+(1/4) (3+Cos[4 θ]) ((Sin[ϕ])^4))==0,r]//Union

Maximize[{(1/((1+((Cos[ϕ])^4)+(1/4) (3+Cos[4 θ]) ((Sin[ϕ])^4)))),0<θ<2π&&0<ϕ<π},{θ,ϕ}]

NIntegrate[Sin[ϑ]/(1+Cos[ϑ]^4+(1/4)*(3+Cos[4*φ])*Sin[ϑ]^4)^(3/4),{φ,0,2*Pi},{ϑ,0,Pi},Method->"GaussKronrod",PrecisionGoal->16,WorkingPrecision->30,MaxRecursion->10]/3

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

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

Integrate[Boole[-1+(r^4)+(r^4) (((Cos[ϕ])^4)+(1/4) (3+Cos[4 θ]) ((Sin[ϕ])^4))<0](r^2)Sin[ϕ],{r,0,2},Assumptions->0<θ<2π&&0<ϕ<π]

NIntegrate[(1/3) ((Root[-4+(4+4 ((Cos[ϕ])^4)+3 ((Sin[ϕ])^4)+Cos[4 θ] ((Sin[ϕ])^4)) ((#1)^4)&,2])^3) Sin[ϕ],{θ,0,2π},{ϕ,0,π}]

Integrate[(1/3) ((Root[-4+(4+4 ((Cos[ϕ])^4)+3 ((Sin[ϕ])^4)+Cos[4 θ] ((Sin[ϕ])^4)) ((#1)^4)&,2])^3) Sin[ϕ],{θ,0,2π},Assumptions->0<ϕ<π]

Integrate[(1/3) ((Root[-4+(4+4 ((Cos[ϕ])^4)+3 ((Sin[ϕ])^4)+Cos[4 θ] ((Sin[ϕ])^4)) ((#1)^4)&,2])^3) Sin[ϕ],{ϕ,0,π},Assumptions->0<θ<2π]

ToRadicals[(1/3) ((Root[-4+(4+4 ((Cos[ϕ])^4)+3 ((Sin[ϕ])^4)+Cos[4 θ] ((Sin[ϕ])^4)) ((#1)^4)&,2])^3) Sin[ϕ],Assumptions->r>0&&0<ϕ<π]

Integrate[Boole[-1+(r^4)+(r^4) (((Cos[ϕ])^4)+(1/4) (3+Cos[4 θ]) ((Sin[ϕ])^4))<0](r^2)Sin[ϕ],{θ,0,2π},Assumptions->0<r<2&&0<ϕ<π]

Integrate[Boole[-1+(r^4)+(r^4) (((Cos[ϕ])^4)+(1/4) (3+Cos[4 θ]) ((Sin[ϕ])^4))<0](r^2)Sin[ϕ],{ϕ,0,π},Assumptions->0<r<2&&0<θ<2π]

{a==0,b==0,c==-1}

RegionPlot3D[Goursat[0,0,-1][x,y,z]<0,{x,-1,1},{y,-1,1},{z,-1,1}]

surf=Goursat[0,0,-1][x,y,z]

NIntegrate[Boole[surf<0],{x,-1,1},{y,-1,1},{z,-1,1},Method->"AdaptiveQuasiMonteCarlo"]

Integrate[Boole[surf<0],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞}]

(∫_(-1)^1)(∫_(Root[-1+(x^4)+((#1)^4)&,1])^(Root[-1+(x^4)+((#1)^4)&,2]))(-Root[-1+(x^4)+(y^4)+((#1)^4)&,1]+Root[-1+(x^4)+(y^4)+((#1)^4)&,2])\[DifferentialD]y\[DifferentialD]x

NIntegrate[-Root[-1 + x^4 + y^4 + #1^4 & , 1] + Root[-1 + x^4 + y^4 + #1^4 & , 2], {x, -1, 1}, 
  {y, Root[-1 + x^4 + #1^4 & , 1], Root[-1 + x^4 + #1^4 & , 2]},Method->"AdaptiveQuasiMonteCarlo"]

ToRadicals[Root[-1+(x^4)+(y^4)+((#1)^4)&,1]]

ToRadicals[Root[-1+(x^4)+(y^4)+((#1)^4)&,2]]

Plot3D[Root[-1+(x^4)+(y^4)+((#1)^4)&,2]-(((1-(x^4)-(y^4)))^(1/4)),{x,-1,1},{y,-1,1}]

ToRadicals[Root[-1+(x^4)+((#1)^4)&,2]]

2NIntegrate[(((1-(x^4)-(y^4)))^(1/4)),{x, -1, 1},{y,-(((1-(x^4)))^(1/4)),(((1-(x^4)))^(1/4))}]

4NIntegrate[(((1-(x^4)-(y^4)))^(1/4)),{x, -1, 1},{y,0,(((1-(x^4)))^(1/4))}]

8NIntegrate[(((1-(x^4)-(y^4)))^(1/4)),{x, 0, 1},{y,0,(((1-(x^4)))^(1/4))}]

8Integrate[(((1-(x^4)-(y^4)))^(1/4)),{x, 0, 1},{y,0,(((1-(x^4)))^(1/4))}]

<<Utilities`Simplify`

GammaSimplify[((8 ((Gamma[(5/4)])^3))/(Gamma[(7/4)]))]

N[%]

8(((Gamma[1+(1/4)])^3)/(Gamma[1+(3/4)]))//N

{a==0,b==-2,c==-1}

surf=Goursat[0,-2,-1][x,y,z]

RegionPlot3D[surf<0,{x,-1.8,1.8},{y,-1.8,1.8},{z,-1.8,1.8}]

NIntegrate[Boole[surf<0],{x,-1.8,1.8},{y,-1.8,1.8},{z,-1.8,1.8},Method->"AdaptiveQuasiMonteCarlo"]

ByteCount/@Integrate[Boole[surf<0],{#1,0,18/10},{#2,0,18/10},{#3,0,18/10}]&@@@Permutations[{x,y,z}]

Integrate[Boole[surf<0],{x,0,18/10},{y,0,18/10},{z,0,18/10}]

{a==0,b==-1,c==(1/2)}

surf=Goursat[0,-1,1/2][x,y,z]

Default Orientation

RegionPlot3D[surf<0,{z,-1,1},{y, -1, 1}, {x, -1, 1}]

RegionPlot3D[surf<0,{z,0,1},{y, 0, 1}, {x, 0, 1}]

Singular Points

pts=SingularPoints[surf,{x,y,z}]//FullSimplify

Show[{RegionPlot3D[surf<0,{z,-1,1},{y, -1, 1}, {x, -1, 1}],Graphics3D[{PointSize[.05],Point[pts]}]}]

tri=Select[pts,And@@Thread[#≥0]&]

ctr=Mean[tri]

Show[{RegionPlot3D[surf<0,{z,-1,1},{y, -1, 1}, {x, -1, 1}],Graphics3D[{PointSize[.05],Point[tri],Red,Point[ctr]}]}]

polys=Polygon[{{0,0,0},#,ctr}]&/@Take[tri,2]

Show[{RegionPlot3D[surf<0,{z,-1,1},{y, -1, 1}, {x, -1, 1}],Graphics3D[{PointSize[.05],Point[tri],Red,Point[ctr],polys}]}]

ineq=PolyhedronInequalities[polys,{x,y,z}]

Show[{RegionPlot3D[surf<0&&And@@ineq,{z,0,1},{y, 0, 1}, {x, 0, 1},PlotPoints->50],Graphics3D[{PointSize[.05],Point[tri],Red,Point[ctr],polys}]}]

8 3 2NIntegrate[Boole[surf<0&&x≤z≤y],{z,0,1},{y, 0, 1}, {x, 0, 1},Method->"AdaptiveQuasiMonteCarlo",MaxPoints->10^6,PrecisionGoal->3]

8 3 2NIntegrate[Boole[surf<0], {y, 0, 1},{z,0, y},{x,0,z},Method->"AdaptiveQuasiMonteCarlo",MaxPoints->10^6,PrecisionGoal->3]//Timing

8 3 2Integrate[Boole[surf<0], {y, 0, 1},{z,0, y},{x,0,z}]

i=48 ((∫_(Root[1-6 ((#1)^2)+6 ((#1)^4)&,3])^(Root[1-6 ((#1)^2)+6 ((#1)^4)&,4]))(∫_(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,3])^y)(z-Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3])\[DifferentialD]z\[DifferentialD]y+(∫_(Root[1-6 ((#1)^2)+6 ((#1)^4)&,4])^(Root[1-8 ((#1)^2)+8 ((#1)^4)&,4]))(∫_(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,3])^(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,4]))(z-Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3])\[DifferentialD]z\[DifferentialD]y+(∫_(Root[1-8 ((#1)^2)+8 ((#1)^4)&,4])^1)(∫_(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,3])^(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,4]))(z-Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3])\[DifferentialD]z\[DifferentialD]y+(∫_(Root[1-6 ((#1)^2)+6 ((#1)^4)&,4])^(Root[1-8 ((#1)^2)+8 ((#1)^4)&,4]))(∫_(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,4])^y)(-Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,4])\[DifferentialD]z\[DifferentialD]y+(∫_(Root[1-8 ((#1)^2)+8 ((#1)^4)&,4])^1)(∫_(Root[1-2 (y^2)+2 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,4])^(Root[1-4 (y^2)+4 (y^4)-4 ((#1)^2)+4 ((#1)^4)&,4]))(-Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,4])\[DifferentialD]z\[DifferentialD]y);

i/.Integrate[x__]:>NIntegrate[x,Method->"AdaptiveQuasiMonteCarlo"]

i/.Integrate:>integrate/.r_Root:>ToRadicals[r]/.integrate->NIntegrate

ToRadicals[Root[1-2 (y^2)+2 (y^4)-2 (z^2)+2 (z^4)-2 ((#1)^2)+2 ((#1)^4)&,3]]

FullSimplify[ToRadicals/@PiecewiseExpand[%236]]

FullSimplify[%236,0<z<y<1]

Numerical

8NIntegrate[Boole[surf<0],{z,0,1},{y, 0, 1}, {x, 0, 1},Method->"AdaptiveQuasiMonteCarlo"]

8NIntegrate[Boole[surf<0],{z,0,1},{y, 0, 1}, {x, 0, 1},Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->5,MaxPoints->10^7]

NIntegrate[Boole[surf<0],{z,-1,1},{y, -1, 1}, {x, -1, 1},Method->"AdaptiveQuasiMonteCarlo"]

Cartesian

Integrate[Boole[surf<0],{#1,0,1},{#2,0,1},{#3,0,1}]&@@@Permutations[{x,y,z}]

8NIntegrate[Boole[surf<0],{#1,0,1},{#2,0,1},{#3,0,1},Method->"AdaptiveQuasiMonteCarlo"]&@@@Permutations[{x,y,z}]

Mean[%]

8 NIntegrate[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4],{x,0,1},{y,Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,4]}]

8Integrate[Boole[surf<0],{x,0,1},{y,0,1},{z,0,1}]

FullSimplify[%]

root=Root[1-4 x^2+4 x^4-4 #1^2+4 #1^4&,4];

FullSimplify[root,Assumptions->0<x<1]

ToRadicals[root,Assumptions->0<x<1]

ByteCount/@{%,root}

Options[NIntegrate]

8 NIntegrate[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4],{x,0,1},{y,Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,4]},Method->{"AdaptiveQuasiMonteCarlo","SymbolicProcessing"->False},MaxPoints->10]

Plot[{Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,4]},{x,0,1}]

ToRadicals[Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,4]]

Plot[(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])-Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,4],{x,0,1}]

8 NIntegrate[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4],{x,0,1},{y,Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])},Method->{"AdaptiveQuasiMonteCarlo","SymbolicProcessing"->False},MaxPoints->10]

ToRadicals[Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3]]

Refine[Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],0<x<1]

Plot[(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])])-Root[1-4 (x^2)+4 (x^4)-4 ((#1)^2)+4 ((#1)^4)&,3],{x,0,1}]

Plot[(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),{x,0,1}]

8 NIntegrate[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4],{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])},Method->{"AdaptiveQuasiMonteCarlo","SymbolicProcessing"->False},MaxPoints->10]

8 NIntegrate[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]+Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4],{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

p1=-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3];

Plot3D[p1,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

ToRadicals[-Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,3]]

p2=-(Sqrt[(1/2)-(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])

Plot3D[p2,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

Plot3D[p1-p2,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

p1=Root[1-2 (x^2)+2 (x^4)-2 (y^2)+2 (y^4)-2 ((#1)^2)+2 ((#1)^4)&,4];

Plot3D[p1,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

ToRadicals[p1]

p2=(Sqrt[(1/2)+(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])]);

Plot3D[p2,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

Plot3D[p1-p2,{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

8 NIntegrate[-(Sqrt[(1/2)-(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])+(Sqrt[(1/2)+(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])]),{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])},Method->{"AdaptiveQuasiMonteCarlo","SymbolicProcessing"->False},MaxPoints->10]

8 NIntegrate[(Sqrt[(1/2)+(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[(1/2)-(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])]),{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])},WorkingPrecision->10]

FullSimplify[(Sqrt[(1/2)+(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[(1/2)-(1/2) (Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])]),0<x<1]

(8/(Sqrt[2]))NIntegrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])},WorkingPrecision->10]

(8/(Sqrt[2]))Integrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]

IntegrateWithAssumptions[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-(Sqrt[(x^2)-(x^4)])]),(Sqrt[(1/2)+(Sqrt[(x^2)-(x^4)])])}]//Timing

(8/(Sqrt[2]))NIntegrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-x(Sqrt[1-(x^2)])]),(Sqrt[(1/2)+x(Sqrt[1-(x^2)])])},WorkingPrecision->20]

(8/(Sqrt[2]))NIntegrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-x(Sqrt[1-(x^2)])]),(Sqrt[(1/2)+x(Sqrt[1-(x^2)])])},Method->"AdaptiveQuasiMonteCarlo",MaxPoints->10^7,PrecisionGoal->5,MaxRecursion->25]

4(Sqrt[2]) (∫_0^1)(∫_(Sqrt[1/2-x (Sqrt[1-(x^2)])])^(Sqrt[(Sqrt[1-(x^2)]) x+1/2]))((Sqrt[(Sqrt[-4 (x^4)+4 (x^2)-4 (y^4)+4 (y^2)-1])+1])-(Sqrt[1-(Sqrt[-4 (x^4)+4 (x^2)-(((1-2 (y^2)))^2)])]))\[DifferentialD]y\[DifferentialD]x

(8/(Sqrt[2]))NIntegrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])])-(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-x(Sqrt[1-(x^2)])]),(Sqrt[(1/2)+x(Sqrt[1-(x^2)])])},Method->"GlobalAdaptive",PrecisionGoal->5,WorkingPrecision->100]

(8/(Sqrt[2]))(NIntegrate[(Sqrt[1+(Sqrt[-1+4 (x^2)-4 (x^4)+4 (y^2)-4 (y^4)])]),{x,0,1},{y,(Sqrt[(1/2)-x(Sqrt[1-(x^2)])]),(Sqrt[(1/2)+x(Sqrt[1-(x^2)])])},WorkingPrecision->20]-NIntegrate[(Sqrt[1-(Sqrt[4 (x^2)-4 (x^4)-(((1-2 (y^2)))^2)])]),{x,0,1},{y,(Sqrt[(1/2)-x(Sqrt[1-(x^2)])]),(Sqrt[(1/2)+x(Sqrt[1-(x^2)])])},WorkingPrecision->20])

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

1-D

Michael Trott, Nov. 9, 2008

-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))//TraditionalForm

FullSimplify[ComplexExpand[Re[-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))],t>0],t>0]

<<Utilities`Simplify`

FullSimplify[Re[GammaSimplify[-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))]],t>0]//TraditionalForm

Re[(((-1))^(5/8))]

NIntegrate[Re[-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))],{t,0,1,Infinity},WorkingPrecision->20,PrecisionGoal->6,Method->"DoubleExponential",MaxRecursion->10]

Plot[Through[{Re,Im}[-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))]],{t,0,100},PlotRange->{-.1,.05},PlotStyle->{Red,Yellow},Evaluated->True]

LogPlot[Evaluate[Abs@Through[{Re,Im}[-(((-1)^(5/8)*Pi^2*((-1)^(1/4)*E^((I*t)/8)*t^(3/4)*(-BesselJ[-(1/4),t/8]+(-1)^(3/4)*BesselJ[1/4,t/8])^3+(8*(-1)^(1/4))/Gamma[3/4]^3+(6*Sqrt[t])/(Gamma[3/4]^2*Gamma[5/4])))/(16*Sqrt[2]*t^(7/4)))]]],{t,0,200},PlotRange->{-.07,.05},PlotStyle->{Red,Yellow}]

Rotated Orientation

mat=RotationMatrixZ[π/4].RotationMatrixY[-ArcTan[(1/(Sqrt[3]))]]

surf1=surf/.Thread[{x,y,z}->mat.{x,y,z}]//FullSimplify

RegionPlot3D[surf1<0&&z>0,{z,-1.6,1.6},{y, -1.6,1.6}, {x, -1.6, 1.6},MaxRecursion->3,Axes->True,AxesOrigin->{0,0,0},PlotPoints->30,AxesLabel->{x,y,z}]

8NIntegrate[Boole[surf1<0],{z,0,1.6},{y,0,1.6}, {x,0, 1.6},Method->"AdaptiveQuasiMonteCarlo"]

8NIntegrate[Boole[surf1<0],{z,0,1.6},{y,0,1.6}, {x,0, 1.6},Method->"AdaptiveQuasiMonteCarlo"]

RegionPlot3D[surf1<0,{z,0,1.6},{y, 0,1.6}, {x, 0, 1.6},MaxRecursion->3,Axes->True,AxesOrigin->{0,0,0},PlotPoints->30,AxesLabel->{x,y,z}]

Solve[surf1==0,z]

surf1/.z->0

Minimize[{x,1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])==0&&x>0},{x,y}]

ContourPlot[(Sqrt[1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),{x,.5,1.4},{y,0,1}]

Plot3D[(Sqrt[1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),{x,(1/(Sqrt[3])),1.6},{y,0,1.6},Filling->0]

Plot3D[(Sqrt[1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),{x,.5,(1/(Sqrt[3]))},{y,0,1.6},PlotRange->All]

Plot3D[(Sqrt[1-3 (x^2)-(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),{x,.5,(1/(Sqrt[3]))},{y,0,1.6},PlotRange->All,PlotPoints->20,WorkingPrecision->50]

Maximize[{(Sqrt[1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),x≥1/2},{x,y}]

NIntegrate[(Sqrt[1-3 (x^2)+(Sqrt[2]) (Sqrt[-2 (x^2)+4 (x^4)+(y^2)-(y^4)])]),{x,.5,1.6},{y,0,1.6},M]

RegionPlot3D[surf1<0&&x<=z,{z,0,1.6},{y, 0,1.6}, {x, 0, 1.6},MaxRecursion->3,Axes->True,AxesOrigin->{0,0,0},PlotPoints->30,AxesLabel->{x,y,z}]

16NIntegrate[Boole[surf1<0&&x<=z],{z,0,1.6},{y,0,1.6}, {x,0, 1.6},Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->3]

i=8Integrate[Boole[surf1<0],{z,0,2},{y,0,2}, {x,0, z}]

i/.Integrate[x__]:>NIntegrate[x,Method->"AdaptiveQuasiMonteCarlo",PrecisionGoal->3]

Spherical

FullSimplify[surf/.{x->r Cos[θ]Sin[ϕ],y->r Sin[θ]Sin[ϕ],z->r Cos[ϕ]},r>0&&0<θ<2π&&0<ϕ<π]

Integrate[Boole[(1/32) (16-32 (r^2)+21 (r^4)+(r^4) (4 Cos[2 ϕ]+7 Cos[4 ϕ]+8 Cos[4 θ] ((Sin[ϕ])^4)))<0](r^2)Sin[ϕ],{θ,0,2π},{ϕ,0,π},{r,0,10}]

Integrate[Boole[(1/32) (16-32 (r^2)+21 (r^4)+(r^4) (4 Cos[2 ϕ]+7 Cos[4 ϕ]+8 Cos[4 θ] ((Sin[ϕ])^4)))<0](r^2)Sin[ϕ],{ϕ,0,π},{θ,0,2π},{r,0,10}]

Cylindrical

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

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

FullSimplify[%,r>0&&z∈Reals]

Integrate[%,{z,-∞,∞},{r,0,∞}]

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