Spheroid

Author

Eric W. Weisstein
June 19, 2013

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

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

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

Parametric

x[θ_,ϕ_]:=a Cos[θ]Sin[ϕ]
y[θ_,ϕ_]:=b Sin[θ]Sin[ϕ]
z[θ_,ϕ_]:=c Cos[ϕ]

Oblate

ParametricPlot3D[{x[θ,ϕ],y[θ,ϕ],z[θ,ϕ]}/.{a->1.3,b->1.3,c->1},{θ,0,2π},{ϕ,0,π},ViewPoint->{0,-4,1},Boxed->False,Axes->False]

Sphere

ParametricPlot3D[{x[θ,ϕ],y[θ,ϕ],z[θ,ϕ]}/.{a->1,b->1,c->1},{θ,0,2π},{ϕ,0,π},ViewPoint->{0,-4,1},Boxed->False,Axes->False]

Prolate

ParametricPlot3D[{x[θ,ϕ],y[θ,ϕ],z[θ,ϕ]}/.{a->1,b->1.3,c->1.3},{θ,0,2π},{ϕ,0,π},ViewPoint->{0,-4,1},Boxed->False,Axes->False]

Equations

Numerator[Factor[(x^2+y^2)/a^2+z^2/c^2-1]]

Surface Area

Via Integration

surface={a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]};

FullSimplify[AreaElement[surface,{u,v}],a>0&&c>0&&0<v<π&&0<u<2π]

2π Integrate[((a (Sqrt[(a^2)+(c^2)+(a-c) (a+c) Cos[2 v]]) Sin[v])/(Sqrt[2])),{v,0,π},Assumptions->0<a<c]

FullSimplify[%13,a>0&&c>0]

2π Integrate[((a (Sqrt[(a^2)+(c^2)+(a-c) (a+c) Cos[2 v]]) Sin[v])/(Sqrt[2])),{v,0,π},Assumptions->0<c<a]

Using Hypergeometric2F1

2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2])

Formua Comparisons

Equal@@@Table[
{
With[{ea=Sqrt[1-c^2/a^2]},2Pi a^2+((Pi c^2)/ea) Log[((1+ea)/(1-ea))]],
With[{ec=Sqrt[1-a^2/c^2]},2Pi a^2+((2Pi a c)/ec) ArcSin[ec]],
2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2]),
 2 a π (a+((c^2)/(Sqrt[(a^2)-(c^2)])) ArcCoth[(a/(Sqrt[(a^2)-(c^2)]))]),
2 π ((a^2)+(((c^2) ArcTanh[(Sqrt[1-((c^2)/(a^2))])])/(Sqrt[1-((c^2)/(a^2))]))),
(1/(Sqrt[-(a^2)+(c^2)]))a π (2 a (Sqrt[-(a^2)+(c^2)])-I (c^2) (Log[(a^2)-(c^2)+I a (Sqrt[-(a^2)+(c^2)])]-Log[-(a^2)+(c^2)+I a (Sqrt[-(a^2)+(c^2)])]))
}/.{a->RandomReal[WorkingPrecision->20],c->RandomReal[WorkingPrecision->20]},{1000}]//Union

{
With[{ea=Sqrt[1-c^2/a^2]},2Pi a^2+((Pi c^2)/ea) Log[((1+ea)/(1-ea))]],
With[{ec=Sqrt[1-a^2/c^2]},2Pi a^2+((2Pi a c)/ec) ArcSin[ec]],
2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2]),
 2 a π (a+((c^2)/(Sqrt[(a^2)-(c^2)])) ArcCoth[(a/(Sqrt[(a^2)-(c^2)]))]),
2 π ((a^2)+(((c^2) ArcTanh[(Sqrt[1-((c^2)/(a^2))])])/(Sqrt[1-((c^2)/(a^2))]))),
(1/(Sqrt[-(a^2)+(c^2)]))a π (2 a (Sqrt[-(a^2)+(c^2)])-I (c^2) (Log[(a^2)-(c^2)+I a (Sqrt[-(a^2)+(c^2)])]-Log[-(a^2)+(c^2)+I a (Sqrt[-(a^2)+(c^2)])]))
}/.{a->1,c->1}

Hypergeometric2F1[1/2,1,3/2,1-a^2/a^2]

ReleaseHold[Hold[Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2]]/.a->c]

2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2])/.a->c

Sin[1-c^2/a^2]/.a->c

Grid[Table[
{
With[{ea=Sqrt[1-c^2/a^2]},2Pi a^2+((Pi c^2)/ea) Log[((1+ea)/(1-ea))]],
With[{ec=Sqrt[1-a^2/c^2]},2Pi a^2+((2Pi a c)/ec) ArcSin[ec]],
 2 a π (a+((c^2)/(Sqrt[(a^2)-(c^2)])) ArcCoth[(a/(Sqrt[(a^2)-(c^2)]))]),(1/(Sqrt[-(a^2)+(c^2)]))a π (2 a (Sqrt[-(a^2)+(c^2)])-I (c^2) (Log[(a^2)-(c^2)+I a (Sqrt[-(a^2)+(c^2)])]-Log[-(a^2)+(c^2)+I a (Sqrt[-(a^2)+(c^2)])])),
2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2])
}/.{a->RandomReal[WorkingPrecision->10],c->RandomReal[WorkingPrecision->10]},{10}],Alignment->Left,Dividers->All]

Grid[Table[
{
With[{ea=Sqrt[1-c^2/a^2]},2Pi a^2+((Pi c^2)/ea) Log[((1+ea)/(1-ea))]],
With[{ec=Sqrt[1-a^2/c^2]},2Pi a^2+((2Pi a c)/ec) ArcSin[ec]],
 2 a π (a+((c^2)/(Sqrt[(a^2)-(c^2)])) ArcCoth[(a/(Sqrt[(a^2)-(c^2)]))]),
(1/(Sqrt[-(a^2)+(c^2)]))a π (2 a (Sqrt[-(a^2)+(c^2)])-I (c^2) (Log[(a^2)-(c^2)+I a (Sqrt[-(a^2)+(c^2)])]-Log[-(a^2)+(c^2)+I a (Sqrt[-(a^2)+(c^2)])])),
2 Pi (a^2 +c^2 Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2])
}/.{a->RandomReal[],c->RandomReal[]},{10}],Alignment->Left,Dividers->All]

x=RandomReal[{0,1},{10,2}]

{#,Hypergeometric2F1[1/2,1,3/2,1-c^2/a^2]/.{a->#[[1]],c->#[[2]]}}&/@x

{#,Hypergeometric2F1[1/2,1,3/2,1-#[[2]]^2/#[[1]]^2]}&/@x

Volume

Direct Integration

Solve[(x^2+y^2)/a^2+z^2/c^2==1,z]

v5.0.1

Integrate[1,{x,-a,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,-c (Sqrt[1-(((x^2)+(y^2))/(a^2))]),c (Sqrt[1-(((x^2)+(y^2))/(a^2))])},Assumptions->0<c<a]//Timing

...wrong...

v5.1.1

Integrate[1,{x,-a,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,-c (Sqrt[1-(((x^2)+(y^2))/(a^2))]),c (Sqrt[1-(((x^2)+(y^2))/(a^2))])},Assumptions->0<c<a]//Timing

v6

Integrate[1,{x,-a,a},{y,-Sqrt[a^2-x^2],Sqrt[a^2-x^2]},{z,-c (Sqrt[1-(((x^2)+(y^2))/(a^2))]),c (Sqrt[1-(((x^2)+(y^2))/(a^2))])},Assumptions->0<c<a]//Timing

Boole

v5.0.1

<<Calculus`Integration`

Integrate[Boole[(x^2+y^2)/a^2+z^2/c^2<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->0<c<a,GenerateConditions->True]//Timing

v5.1.1

Integrate[Boole[(x^2+y^2)/a^2+z^2/c^2<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->0<c<a,GenerateConditions->True]//Timing

v6

Integrate[Boole[(x^2+y^2)/a^2+z^2/c^2<1],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&c>0]//Timing

Moments of Inertia

With[{ρ=(M/((4/3)π (a^2) c))},ρ Integrate[{{y^2+z^2,-x y,-x z},
{-x y,z^2+x^2,-y z},
{-x z,-y z,x^2+y^2}}Boole[(((x^2)+(y^2))/(a^2))+((z^2)/(c^2))<1],
{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},Assumptions->a>0&&c>0]
]//TraditionalForm

Metric Properties

Metric Tensor

MetricTensor[{a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]},{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[{a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]},{u,v}]]

TraditionalForm[%]

Surface Properties

Spheroid[a_,c_]:={a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]}

First Fundamental Form

{ℰ,ℱ,𝒢}=FullSimplify[FirstFundamentalForm[Spheroid[a,c],{u,v}],a>0&&c>0]

Second Fundamental Form

FullSimplify[SecondFundamentalForm[Spheroid[a,c],{u,v}],a>0&&c>0&&0<v<π]

Principal Curvatures

FullSimplify[PrincipalCurvatures[Spheroid[a,c],{u,v}],a>0&&c>0&&(u|v)∈Reals]

Gaussian Curvature

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

GaussianCurvature[-(a^2) (c^2)+(c^2) (x^2)+(c^2) (y^2)+(a^2) (z^2),{x,y,z}]

Mean Curvature

FullSimplify[MeanCurvature[Spheroid[a,c],{u,v}],a>0&&c>0&&0<v<π]

Vector Properties

Normal Vector

Parametric

FullSimplify[NormalVector[{a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]},{u,v}],a>0&&c>0&&0<u<2π&&0<v<π]

Implicit

FullSimplify[NormalVector[(x^2+y^2)/a^2+z^2/c^2==1,{x,y,z}],a>0&&c>0&&(x|y|z)∈Reals]

Vector Length

FullSimplify[Norm[{a Sin[v]Cos[u],a Sin[v]Sin[u],c Cos[v]}],a>0&&c>0&&0<u<2π&&0<v<π]/.Abs[x_]^2:>x^2