Astroidal Ellipsoid

Author

Eric W. Weisstein
June 20, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Initialization

AstroidalEllipse[u_,v_]:={a,b,c}{Cos[u]Cos[v],Sin[u]Cos[v],Sin[v]}^3

Special Case

AstroidalEllipse[u,v]/.Thread[{a,b,c}->{1,1,1}]

hyperbolic octahedron

Graphics

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

V5

Show[GraphicsArray[Block[{$DisplayFunction=Identity},ParametricPlot3D[AstroidalEllipse[u,v]/.Thread[{a,b,c}->#],{u,-π/2,π/2},{v,-π,π},PlotRange->All,Axes->False,Boxed->False,PlotLabel->TraditionalForm[label@@#]]&/@{{1,1,1},{1,1,2},{1,2,3}}
],GraphicsSpacing->-.3],ImageSize->400]

V6

GraphicsGrid[{ParametricPlot3D[AstroidalEllipse[u,v]/.Thread[{a,b,c}->#],{u,-π/2,π/2},{v,-π,π},PlotRange->All,Axes->False,Boxed->False,PlotLabel->label@@#]&/@{{1,1,1},{1,1,2},{1,2,3}}},ImageSize->500,Spacings->-100]

Cartesian Equation

{x,y,z}=AstroidalEllipse[u,v]

Total[Simplify[(({x,y,z}/{a,b,c})^2)^(1/3),(u|v)∈Reals]]//Simplify

Factor/@GroebnerBasis[{x==a ((Cos[u])^3)((Cos[v])^3),y==b ((Sin[u])^3)((Cos[v])^3),z==c ((Sin[v])^3),((Sin[u])^2)+((Cos[u])^2)==1,((Sin[v])^2)+((Cos[v])^2)==1},{x,y,z},{Sin[u],Cos[u],Sin[v],Cos[v]},MonomialOrder->EliminationOrder]

eqn=-(a^18) (b^18) (c^18)+9 (a^16) (b^18) (c^18) (x^2)-36 (a^14) (b^18) (c^18) (x^4)+84 (a^12) (b^18) (c^18) (x^6)-126 (a^10) (b^18) (c^18) (x^8)+126 (a^8) (b^18) (c^18) (x^10)-84 (a^6) (b^18) (c^18) (x^12)+36 (a^4) (b^18) (c^18) (x^14)-9 (a^2) (b^18) (c^18) (x^16)+(b^18) (c^18) (x^18)+9 (a^18) (b^16) (c^18) (y^2)+9 (a^16) (b^16) (c^18) (x^2) (y^2)-234 (a^14) (b^16) (c^18) (x^4) (y^2)+711 (a^12) (b^16) (c^18) (x^6) (y^2)-990 (a^10) (b^16) (c^18) (x^8) (y^2)+711 (a^8) (b^16) (c^18) (x^10) (y^2)-234 (a^6) (b^16) (c^18) (x^12) (y^2)+9 (a^4) (b^16) (c^18) (x^14) (y^2)+9 (a^2) (b^16) (c^18) (x^16) (y^2)-36 (a^18) (b^14) (c^18) (y^4)-234 (a^16) (b^14) (c^18) (x^2) (y^4)-513 (a^14) (b^14) (c^18) (x^4) (y^4)+2961 (a^12) (b^14) (c^18) (x^6) (y^4)-2961 (a^10) (b^14) (c^18) (x^8) (y^4)+513 (a^8) (b^14) (c^18) (x^10) (y^4)+234 (a^6) (b^14) (c^18) (x^12) (y^4)+36 (a^4) (b^14) (c^18) (x^14) (y^4)+84 (a^18) (b^12) (c^18) (y^6)+711 (a^16) (b^12) (c^18) (x^2) (y^6)+2961 (a^14) (b^12) (c^18) (x^4) (y^6)+12171 (a^12) (b^12) (c^18) (x^6) (y^6)+2961 (a^10) (b^12) (c^18) (x^8) (y^6)+711 (a^8) (b^12) (c^18) (x^10) (y^6)+84 (a^6) (b^12) (c^18) (x^12) (y^6)-126 (a^18) (b^10) (c^18) (y^8)-990 (a^16) (b^10) (c^18) (x^2) (y^8)-2961 (a^14) (b^10) (c^18) (x^4) (y^8)+2961 (a^12) (b^10) (c^18) (x^6) (y^8)+990 (a^10) (b^10) (c^18) (x^8) (y^8)+126 (a^8) (b^10) (c^18) (x^10) (y^8)+126 (a^18) (b^8) (c^18) (y^10)+711 (a^16) (b^8) (c^18) (x^2) (y^10)+513 (a^14) (b^8) (c^18) (x^4) (y^10)+711 (a^12) (b^8) (c^18) (x^6) (y^10)+126 (a^10) (b^8) (c^18) (x^8) (y^10)-84 (a^18) (b^6) (c^18) (y^12)-234 (a^16) (b^6) (c^18) (x^2) (y^12)+234 (a^14) (b^6) (c^18) (x^4) (y^12)+84 (a^12) (b^6) (c^18) (x^6) (y^12)+36 (a^18) (b^4) (c^18) (y^14)+9 (a^16) (b^4) (c^18) (x^2) (y^14)+36 (a^14) (b^4) (c^18) (x^4) (y^14)-9 (a^18) (b^2) (c^18) (y^16)+9 (a^16) (b^2) (c^18) (x^2) (y^16)+(a^18) (c^18) (y^18)+9 (a^18) (b^18) (c^16) (z^2)+9 (a^16) (b^18) (c^16) (x^2) (z^2)-234 (a^14) (b^18) (c^16) (x^4) (z^2)+711 (a^12) (b^18) (c^16) (x^6) (z^2)-990 (a^10) (b^18) (c^16) (x^8) (z^2)+711 (a^8) (b^18) (c^16) (x^10) (z^2)-234 (a^6) (b^18) (c^16) (x^12) (z^2)+9 (a^4) (b^18) (c^16) (x^14) (z^2)+9 (a^2) (b^18) (c^16) (x^16) (z^2)+9 (a^18) (b^16) (c^16) (y^2) (z^2)+3339 (a^16) (b^16) (c^16) (x^2) (y^2) (z^2)-1512 (a^14) (b^16) (c^16) (x^4) (y^2) (z^2)-12222 (a^12) (b^16) (c^16) (x^6) (y^2) (z^2)+12222 (a^10) (b^16) (c^16) (x^8) (y^2) (z^2)+1512 (a^8) (b^16) (c^16) (x^10) (y^2) (z^2)-3339 (a^6) (b^16) (c^16) (x^12) (y^2) (z^2)-9 (a^4) (b^16) (c^16) (x^14) (y^2) (z^2)-234 (a^18) (b^14) (c^16) (y^4) (z^2)-1512 (a^16) (b^14) (c^16) (x^2) (y^4) (z^2)+27837 (a^14) (b^14) (c^16) (x^4) (y^4) (z^2)+6867 (a^12) (b^14) (c^16) (x^6) (y^4) (z^2)+27837 (a^10) (b^14) (c^16) (x^8) (y^4) (z^2)-1512 (a^8) (b^14) (c^16) (x^10) (y^4) (z^2)-234 (a^6) (b^14) (c^16) (x^12) (y^4) (z^2)+711 (a^18) (b^12) (c^16) (y^6) (z^2)-12222 (a^16) (b^12) (c^16) (x^2) (y^6) (z^2)+6867 (a^14) (b^12) (c^16) (x^4) (y^6) (z^2)-6867 (a^12) (b^12) (c^16) (x^6) (y^6) (z^2)+12222 (a^10) (b^12) (c^16) (x^8) (y^6) (z^2)-711 (a^8) (b^12) (c^16) (x^10) (y^6) (z^2)-990 (a^18) (b^10) (c^16) (y^8) (z^2)+12222 (a^16) (b^10) (c^16) (x^2) (y^8) (z^2)+27837 (a^14) (b^10) (c^16) (x^4) (y^8) (z^2)+12222 (a^12) (b^10) (c^16) (x^6) (y^8) (z^2)-990 (a^10) (b^10) (c^16) (x^8) (y^8) (z^2)+711 (a^18) (b^8) (c^16) (y^10) (z^2)+1512 (a^16) (b^8) (c^16) (x^2) (y^10) (z^2)-1512 (a^14) (b^8) (c^16) (x^4) (y^10) (z^2)-711 (a^12) (b^8) (c^16) (x^6) (y^10) (z^2)-234 (a^18) (b^6) (c^16) (y^12) (z^2)-3339 (a^16) (b^6) (c^16) (x^2) (y^12) (z^2)-234 (a^14) (b^6) (c^16) (x^4) (y^12) (z^2)+9 (a^18) (b^4) (c^16) (y^14) (z^2)-9 (a^16) (b^4) (c^16) (x^2) (y^14) (z^2)+9 (a^18) (b^2) (c^16) (y^16) (z^2)-36 (a^18) (b^18) (c^14) (z^4)-234 (a^16) (b^18) (c^14) (x^2) (z^4)-513 (a^14) (b^18) (c^14) (x^4) (z^4)+2961 (a^12) (b^18) (c^14) (x^6) (z^4)-2961 (a^10) (b^18) (c^14) (x^8) (z^4)+513 (a^8) (b^18) (c^14) (x^10) (z^4)+234 (a^6) (b^18) (c^14) (x^12) (z^4)+36 (a^4) (b^18) (c^14) (x^14) (z^4)-234 (a^18) (b^16) (c^14) (y^2) (z^4)-1512 (a^16) (b^16) (c^14) (x^2) (y^2) (z^4)+27837 (a^14) (b^16) (c^14) (x^4) (y^2) (z^4)+6867 (a^12) (b^16) (c^14) (x^6) (y^2) (z^4)+27837 (a^10) (b^16) (c^14) (x^8) (y^2) (z^4)-1512 (a^8) (b^16) (c^14) (x^10) (y^2) (z^4)-234 (a^6) (b^16) (c^14) (x^12) (y^2) (z^4)-513 (a^18) (b^14) (c^14) (y^4) (z^4)+27837 (a^16) (b^14) (c^14) (x^2) (y^4) (z^4)-120312 (a^14) (b^14) (c^14) (x^4) (y^4) (z^4)+120312 (a^12) (b^14) (c^14) (x^6) (y^4) (z^4)-27837 (a^10) (b^14) (c^14) (x^8) (y^4) (z^4)+513 (a^8) (b^14) (c^14) (x^10) (y^4) (z^4)+2961 (a^18) (b^12) (c^14) (y^6) (z^4)+6867 (a^16) (b^12) (c^14) (x^2) (y^6) (z^4)+120312 (a^14) (b^12) (c^14) (x^4) (y^6) (z^4)+6867 (a^12) (b^12) (c^14) (x^6) (y^6) (z^4)+2961 (a^10) (b^12) (c^14) (x^8) (y^6) (z^4)-2961 (a^18) (b^10) (c^14) (y^8) (z^4)+27837 (a^16) (b^10) (c^14) (x^2) (y^8) (z^4)-27837 (a^14) (b^10) (c^14) (x^4) (y^8) (z^4)+2961 (a^12) (b^10) (c^14) (x^6) (y^8) (z^4)+513 (a^18) (b^8) (c^14) (y^10) (z^4)-1512 (a^16) (b^8) (c^14) (x^2) (y^10) (z^4)+513 (a^14) (b^8) (c^14) (x^4) (y^10) (z^4)+234 (a^18) (b^6) (c^14) (y^12) (z^4)-234 (a^16) (b^6) (c^14) (x^2) (y^12) (z^4)+36 (a^18) (b^4) (c^14) (y^14) (z^4)+84 (a^18) (b^18) (c^12) (z^6)+711 (a^16) (b^18) (c^12) (x^2) (z^6)+2961 (a^14) (b^18) (c^12) (x^4) (z^6)+12171 (a^12) (b^18) (c^12) (x^6) (z^6)+2961 (a^10) (b^18) (c^12) (x^8) (z^6)+711 (a^8) (b^18) (c^12) (x^10) (z^6)+84 (a^6) (b^18) (c^12) (x^12) (z^6)+711 (a^18) (b^16) (c^12) (y^2) (z^6)-12222 (a^16) (b^16) (c^12) (x^2) (y^2) (z^6)+6867 (a^14) (b^16) (c^12) (x^4) (y^2) (z^6)-6867 (a^12) (b^16) (c^12) (x^6) (y^2) (z^6)+12222 (a^10) (b^16) (c^12) (x^8) (y^2) (z^6)-711 (a^8) (b^16) (c^12) (x^10) (y^2) (z^6)+2961 (a^18) (b^14) (c^12) (y^4) (z^6)+6867 (a^16) (b^14) (c^12) (x^2) (y^4) (z^6)+120312 (a^14) (b^14) (c^12) (x^4) (y^4) (z^6)+6867 (a^12) (b^14) (c^12) (x^6) (y^4) (z^6)+2961 (a^10) (b^14) (c^12) (x^8) (y^4) (z^6)+12171 (a^18) (b^12) (c^12) (y^6) (z^6)-6867 (a^16) (b^12) (c^12) (x^2) (y^6) (z^6)+6867 (a^14) (b^12) (c^12) (x^4) (y^6) (z^6)-12171 (a^12) (b^12) (c^12) (x^6) (y^6) (z^6)+2961 (a^18) (b^10) (c^12) (y^8) (z^6)+12222 (a^16) (b^10) (c^12) (x^2) (y^8) (z^6)+2961 (a^14) (b^10) (c^12) (x^4) (y^8) (z^6)+711 (a^18) (b^8) (c^12) (y^10) (z^6)-711 (a^16) (b^8) (c^12) (x^2) (y^10) (z^6)+84 (a^18) (b^6) (c^12) (y^12) (z^6)-126 (a^18) (b^18) (c^10) (z^8)-990 (a^16) (b^18) (c^10) (x^2) (z^8)-2961 (a^14) (b^18) (c^10) (x^4) (z^8)+2961 (a^12) (b^18) (c^10) (x^6) (z^8)+990 (a^10) (b^18) (c^10) (x^8) (z^8)+126 (a^8) (b^18) (c^10) (x^10) (z^8)-990 (a^18) (b^16) (c^10) (y^2) (z^8)+12222 (a^16) (b^16) (c^10) (x^2) (y^2) (z^8)+27837 (a^14) (b^16) (c^10) (x^4) (y^2) (z^8)+12222 (a^12) (b^16) (c^10) (x^6) (y^2) (z^8)-990 (a^10) (b^16) (c^10) (x^8) (y^2) (z^8)-2961 (a^18) (b^14) (c^10) (y^4) (z^8)+27837 (a^16) (b^14) (c^10) (x^2) (y^4) (z^8)-27837 (a^14) (b^14) (c^10) (x^4) (y^4) (z^8)+2961 (a^12) (b^14) (c^10) (x^6) (y^4) (z^8)+2961 (a^18) (b^12) (c^10) (y^6) (z^8)+12222 (a^16) (b^12) (c^10) (x^2) (y^6) (z^8)+2961 (a^14) (b^12) (c^10) (x^4) (y^6) (z^8)+990 (a^18) (b^10) (c^10) (y^8) (z^8)-990 (a^16) (b^10) (c^10) (x^2) (y^8) (z^8)+126 (a^18) (b^8) (c^10) (y^10) (z^8)+126 (a^18) (b^18) (c^8) (z^10)+711 (a^16) (b^18) (c^8) (x^2) (z^10)+513 (a^14) (b^18) (c^8) (x^4) (z^10)+711 (a^12) (b^18) (c^8) (x^6) (z^10)+126 (a^10) (b^18) (c^8) (x^8) (z^10)+711 (a^18) (b^16) (c^8) (y^2) (z^10)+1512 (a^16) (b^16) (c^8) (x^2) (y^2) (z^10)-1512 (a^14) (b^16) (c^8) (x^4) (y^2) (z^10)-711 (a^12) (b^16) (c^8) (x^6) (y^2) (z^10)+513 (a^18) (b^14) (c^8) (y^4) (z^10)-1512 (a^16) (b^14) (c^8) (x^2) (y^4) (z^10)+513 (a^14) (b^14) (c^8) (x^4) (y^4) (z^10)+711 (a^18) (b^12) (c^8) (y^6) (z^10)-711 (a^16) (b^12) (c^8) (x^2) (y^6) (z^10)+126 (a^18) (b^10) (c^8) (y^8) (z^10)-84 (a^18) (b^18) (c^6) (z^12)-234 (a^16) (b^18) (c^6) (x^2) (z^12)+234 (a^14) (b^18) (c^6) (x^4) (z^12)+84 (a^12) (b^18) (c^6) (x^6) (z^12)-234 (a^18) (b^16) (c^6) (y^2) (z^12)-3339 (a^16) (b^16) (c^6) (x^2) (y^2) (z^12)-234 (a^14) (b^16) (c^6) (x^4) (y^2) (z^12)+234 (a^18) (b^14) (c^6) (y^4) (z^12)-234 (a^16) (b^14) (c^6) (x^2) (y^4) (z^12)+84 (a^18) (b^12) (c^6) (y^6) (z^12)+36 (a^18) (b^18) (c^4) (z^14)+9 (a^16) (b^18) (c^4) (x^2) (z^14)+36 (a^14) (b^18) (c^4) (x^4) (z^14)+9 (a^18) (b^16) (c^4) (y^2) (z^14)-9 (a^16) (b^16) (c^4) (x^2) (y^2) (z^14)+36 (a^18) (b^14) (c^4) (y^4) (z^14)-9 (a^18) (b^18) (c^2) (z^16)+9 (a^16) (b^18) (c^2) (x^2) (z^16)+9 (a^18) (b^16) (c^2) (y^2) (z^16)+(a^18) (b^18) (z^18);

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

Fundamental Forms

First Fundamental Form

FullSimplify[FirstFundamentalForm[AstroidalEllipse[u,v],{u,v}]]

%//ColumnForm//TraditionalForm

Second Fundamental Form

FullSimplify[SecondFundamentalForm[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]

%//ColumnForm//TraditionalForm

Curvatures

Gaussian Curvature

FullSimplify[GaussianCurvature[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]

%//TraditionalForm

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

k=Last[%];

ByteCount/@k

Mean Curvature

FullSimplify[MeanCurvature[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]

Simplify[MeanCurvature[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]

%//TraditionalForm

Principal Curvatures

Simplify[PrincipalCurvatures[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]

FullSimplify[%5,a>0&&b>0&&c>0&&(u|v)∈Reals]

Area Element

FullSimplify[AreaElement[AstroidalEllipse[u,v],{u,v}],a>0&&b>0&&c>0]//TraditionalForm

FullSimplify[AreaElement[AstroidalEllipse[u,v],{u,v}],-π/2≤u≤π/2&&-π≤v≤π&&a>0&&b>0&&c>0]//Timing

Integrate[9 Abs[Sin[u]] Abs[Sin[v]] Cos[u] ((Cos[v])^4) (Sqrt[(b^2) (c^2) ((Sin[u])^2) ((Sin[v])^2)+(a^2) ((Cos[u])^2) ((b^2) ((Cos[v])^2) ((Sin[u])^2)+(c^2) ((Sin[v])^2))]),{u,-π/2,π/2},{v,-π,π},Assumptions->a>0&&b>0&&c>0]//Timing

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[AstroidalEllipse[u,v],{u,v}],-π/2<u<π/2&&0<v<π&&a>0&&b>0&&c>0]

Vector Properties

eqn={a,b,c}{Cos[u]Cos[v],Sin[u]Cos[v],Sin[v]}^3;

Vector Length

FullSimplify[Norm[eqn],a>0&&b>0&&c>0]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n

Normal Vector

Assuming[a>0&&b>0&&c>0&&-Pi/2<u<Pi/2&&-Pi<v<Pi,FullSimplify[NormalVector[eqn,{u,v}]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n]]//Timing