Boy Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Definition

boy[a_][u_,v_]:=a{(((Sqrt[2])Cos[v]^2 Cos[2u]+Cos[u]Sin[2v])/(2-(Sqrt[2])Sin[3u]Sin[2v])),(((Sqrt[2])Cos[v]^2Sin[2u]+Cos[u]Sin[2v])/(2-(Sqrt[2])Sin[3u]Sin[2v])),((3Cos[v]^2)/(2-(Sqrt[2])Sin[3u]Sin[2v]))}

Equations

eqns=Join[Numerator[Factor[TrigExpand[{x,y,z}-boy[u,v]]]],Cos[#]^2+Sin[#]^2-1&/@{u,v}]

{eqn}=Factor/@GroebnerBasis[eqns,{x,y,z},{Sin[u],Cos[u],Sin[v],Cos[v]},MonomialOrder->EliminationOrder]

eqn=729 (x^6)-2916 (x^5) y+5103 (x^4) (y^2)-5832 (x^3) (y^3)+5103 (x^2) (y^4)-2916 x (y^5)+729 (y^6)+486 (Sqrt[2]) (x^5) z-486 a (x^3) y z-972 (Sqrt[2]) (x^4) y z+1458 a (x^2) (y^2) z+972 (Sqrt[2]) (x^3) (y^2) z-1458 a x (y^3) z-972 (Sqrt[2]) (x^2) (y^3) z+486 a (y^4) z+486 (Sqrt[2]) x (y^4) z+81 (a^2) (x^2) (z^2)-162 (x^4) (z^2)-162 (a^2) x y (z^2)-324 (Sqrt[2]) a (x^2) y (z^2)+1296 (x^3) y (z^2)+81 (a^2) (y^2) (z^2)+486 (Sqrt[2]) a x (y^2) (z^2)-2106 (x^2) (y^2) (z^2)-162 (Sqrt[2]) a (y^3) (z^2)+1944 x (y^3) (z^2)-648 (y^4) (z^2)+54 (Sqrt[2]) (a^2) x (z^3)-216 a (x^2) (z^3)-216 (Sqrt[2]) (x^3) (z^3)-54 (Sqrt[2]) (a^2) y (z^3)+216 a x y (z^3)+432 (Sqrt[2]) (x^2) y (z^3)-216 a (y^2) (z^3)-540 (Sqrt[2]) x (y^2) (z^3)+108 (Sqrt[2]) (y^3) (z^3)+18 (a^2) (z^4)-144 (Sqrt[2]) a x (z^4)+72 (x^2) (z^4)+108 (Sqrt[2]) a y (z^4)-144 x y (z^4)+108 (y^2) (z^4)-48 a (z^5)+96 (Sqrt[2]) x (z^5)-48 (Sqrt[2]) y (z^5)+32 (z^6);

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

Parametrization 1

BoySurface[t_,p_]:=With[
{
x=Cos[t]Sin[p],
y=Sin[t]Sin[p],
z=Cos[p]
},
{
((2x^2-y^2-z^2)(x^2+y^2+z^2)+2y z*(y^2-z^2)+z x*(x^2-z^2)+x y*(y^2-x^2))/2,
 Sqrt[3]/2((y^2-z^2)(x^2+y^2+z^2)+z x*(z^2-x^2)+x y*(y^2-x^2)),
 (x+y+z)((x+y+z)^3+4(y-x)(z-y)(x-z))/8
}
]

BoySurface[u,v]//FullSimplify//TraditionalForm

bs=ParametricPlot3D[Evaluate[BoySurface[t,p]],
  {t,0,Pi},{p,0,Pi},Boxed->False,Axes->False]

Properties

Curvatures[BoySurface[u,v],{u,v}]

Nordstrand's Parametrization 2

V5

ParametricPlot3D[boy,{u,-π/2,π/2},{v,0,π},Boxed->False,Axes->False,PlotRange->All,BoxRatios->{1,1,1}]

V6

ParametricPlot3D[boy[1][u,v],{u,-π/2,π/2},{v,0,π},Boxed->False,Axes->False,BoxRatios->{1,1,1},PlotRange->All]

R. Bryant's Parameterization

gs={ -(3/2)Im[(( z(1-(z^4)))/((z^6)+ (Sqrt[5])(z^3) -1))],-(3/2)Re[ (( z(1+(z^4)))/((z^6) + (Sqrt[5])(z^3) -1))],Im[ ((1+(z^6))/((z^6)+ (Sqrt[5])(z^3) -1))]- (1/2)};

f=FullSimplify[gs/gs.gs/.z->r Exp[I θ],0<r<1&&0<θ<2π]

ParametricPlot3D[Evaluate[f],{r,0,1},{θ,0,2π},PlotPoints->{50,100},Boxed->False,Axes->False]

Homotopy

BoyRoman[alpha_]:={(Sqrt[2]Cos[2u]Cos[v]^2+Cos[u]Sin[2v])/(2-alpha Sqrt[2]Sin[3u]Sin[2v]),(Sqrt[2]Sin[2u]Cos[v]^2-Sin[u]Sin[2v])/(2-alpha Sqrt[2]Sin[3u]Sin[2v]),3Cos[v]^2/(2-alpha Sqrt[2]Sin[3u]Sin[2v])}

Show[GraphicsArray[{
bs,
 Show[bs,ViewPoint->{0,0,2},DisplayFunction->Identity],Show[bs,ViewPoint->{2,-2,0},DisplayFunction->Identity]
}]]

BoyRoman[alpha_]:={
 (Sqrt[2]Cos[2u]Cos[v]^2+Cos[u]Sin[2v])/(2-alpha Sqrt[2]Sin[3u]Sin[2v]),
 (Sqrt[2]Sin[2u]Cos[v]^2-Sin[u]Sin[2v])/(2-alpha Sqrt[2]Sin[3u]Sin[2v]),
 3Cos[v]^2/(2-alpha Sqrt[2]Sin[3u]Sin[2v])}

boy=ParametricPlot3D[BoyRoman[1],{u,-Pi/2,Pi/2},{v,0,Pi},Boxed->False,
  Axes->False,PlotPoints->{30,30}]

Show[GraphicsArray[{boy,Show[boy,ViewPoint->{2,1,2},DisplayFunction->Identity,
				SphericalRegion->True],
 Show[boy,ViewPoint->{0,0,2},DisplayFunction->Identity]}],DisplayFunction->$DisplayFunction]

Show[GraphicsArray[Partition[Table[ParametricPlot3D[Evaluate[BoyRoman[a]],{u,-Pi/2,Pi/2},{v,0,Pi},Boxed->False,Axes->False,PlotPoints->{20,20},DisplayFunction->Identity,SphericalRegion->True],{a,0,1,.2}],3]],DisplayFunction->$DisplayFunction]

b=1;
ParametricPlot3D[(1-Sqrt[2.] t b Sin[3. th]+t^2)^(-1)*{Sqrt[2.]/3. (Cos[2. th]+Sqrt[2.] t Cos[th]),Sqrt[2.]/3. (Sin[2. th]-Sqrt[2.] t Sin[th]),1.},{t,-Infinity,Infinity},{th,0.,2. Pi},Boxed->False,Axes->False,DisplayFunction->Identity]

Simplify[Solve[64(1-z)^3 z^3-48(1-z)^2 z^2(3 x^2+3 y^2+2 z^2)+12(1-z)z (27(x^2+y^2)^2-24 z^2(x^2+y^2)+36 Sqrt[2] y z(y^2-3 x^2)+z^4)+(9 x^2+9 y^2-2 z^2)(-81(x^2+y^2)^2-72 z^2(x^2+y^2)+108 Sqrt[2] x z(x^2-3 y^2)+4 z^4)==0,z]]

Apéry's Inversion

f[θ_,τ_]:=1/(1+τ^2-τ (Sin[2θ]-Cos[2θ]))*{(1+2τ) Cos[θ]/Sqrt[3],(1-2τ) Sin[θ]/Sqrt[3],1};

Factor/@GroebnerBasis[Append[TrigExpand[Numerator[Together[{x,y,z}-f[θ,τ]]]],Cos[θ]^2+Sin[θ]^2-1],{x,y,z},{τ,Cos[θ],Sin[θ]},MonomialOrder->EliminationOrder]//Timing

Properties

Curvatures[boy[a][u,v],{u,v},Simplify[#,a>0]&]

Curvatures[boy[a][u,v],{u,v},FullSimplify[Simplify[#,a>0]]&]

GaussianCurvature[eqn,{x,y,z},Identity]//Timing

ByteCount/@Last[%]

FullSimplify[Last[%4]]

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ByteCount/@k

k[[2]]

Fundamental Forms

Simplify[FirstFundamentalForm[boy[u,v],{u,v}]]

FullSimplify[FirstFundamentalForm[boy[u,v],{u,v}],-π/2≤u≤π/2&&0≤v≤π]

%//ColumnForm//TraditionalForm

Simplify[SecondFundamentalForm[boy[u,v],{u,v}]]

FullSimplify[%,-π/2≤u≤π/2&&0≤v≤π]

%//ColumnForm//TraditionalForm

Curvatures

Gaussian

Simplify[GaussianCurvature[boy[a][u,v],{u,v}],-π/2≤u≤π/2&&0≤v≤π]

...ran overnight...

Mean

Simplify[MeanCurvature[boy[a][u,v],{u,v}],-π/2≤u≤π/2&&0≤v≤π]

Principal

Simplify[PrincipalCurvatures[boy[a][u,v],{u,v}],-π/2≤u≤π/2&&0≤v≤π]

Metric Properties

Metric Tensor

FullSimplify[MetricTensor[boy[a][u,v],{u,v}],-π/2<u<π/2&&0<v<π]

...ran overnight...

Simplify[MetricTensor[boy[a][u,v],{u,v}],-π/2<u<π/2&&0<v<π]

FullSimplify[%,-π/2<u<π/2&&0<v<π]

...ran overnight...

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[boy[a][u,v],{u,v}],-π/2<u<π/2&&0<v<π]

...ran overnight...

Simplify[ChristoffelSymbols[boy[a][u,v],{u,v}],-π/2<u<π/2&&0<v<π]

Vector Properties

eqn=a{(((Sqrt[2])Cos[v]^2 Cos[2u]+Cos[u]Sin[2v])/(2-(Sqrt[2])Sin[3u]Sin[2v])),(((Sqrt[2])Cos[v]^2Sin[2u]+Cos[u]Sin[2v])/(2-(Sqrt[2])Sin[3u]Sin[2v])),((3Cos[v]^2)/(2-(Sqrt[2])Sin[3u]Sin[2v]))};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&-Pi/2<u<π/2&&0<v<π;

Vector Length

Assuming[assum,Simplify[Norm[eqn]/.rules]]//Timing

Assuming[assum,FullSimplify[(a √(11 ((Cos[v])^4)+2 (Sqrt[2]) ((Cos[v])^3) (Cos[u]+Cos[3 u]+Sin[u]+Sin[3 u]) Sin[v]+2 ((Cos[u])^2) ((Sin[2 v])^2)))/(2-(Sqrt[2]) Sin[3 u] Sin[2 v])]]//Timing

Normal Vector

Assuming[assum,Simplify[NormalVector[eqn,{u,v}]/.rules]]//Timing