Prolate Spheroid

Author

Eric W. Weisstein
April 11, 2006

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SolidGeometry/ProlateSpheroid.nb.

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

©2006 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Graphic

Show[Graphics3D[Spheroid[0.8,1]],Boxed->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm[prospher.m,Show[Graphics3D[Spheroid[0.8,1]],opts]]

Parametric

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

((#)/((a^2)(c^2)))&/@GroebnerBasis[{x-a cosu sinv,y-a sinu sinv,z-c cosv,sinu^2+cosu^2-1,sinv^2+cosv^2-1},{x,y,z},{sinu,cosu,sinv,cosv},MonomialOrder->EliminationOrder][[1]]//Simplify

Fundamental Forms

FirstFundamentalForm[{a Cos[u]Sin[v],a Sin[u]Sin[v],c Cos[v]},{u,v}]//Simplify//TraditionalForm

Simplify[SecondFundamentalForm[{a Cos[u]Sin[v],a Sin[u]Sin[v],c Cos[v]},{u,v}],a>0]//TraditionalForm

Volume

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

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

Centroid

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

(((1/4) (a^3) c π)/((4/3) (a^2) c π))

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

(((1/4) (a^2) (c^2) π)/((4/3) (a^2) c π))

Surface Area

Radial

Expand[FullSimplify[2Pi a Integrate[Sqrt[1+(a-c)(a+c)z^2/c^4],{z,-c,c},Assumptions->0<a<c]
]]

With[{a=1,c=3},2 (a^2) π+((2 a (c^2) π ArcCsc[(c/(Sqrt[-(a^2)+(c^2)]))])/(Sqrt[-(a^2)+(c^2)]))]//N

Via First Fundamental Form

FullSimplify[(Sqrt[ℰ 𝒢-(ℱ^2)]),0<a<c&&0<u<2π&&0<v<π]

With[{a=1,c=3},2π NIntegrate[((a (Sqrt[(a^2)+(c^2)+(a-c) (a+c) Cos[2 v]]) Sin[v])/(Sqrt[2])),{v,0,π}]]

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]//Timing

s=FullSimplify[((a π (2 (Sqrt[-(a^4)+(a^2) (c^2)])+I (c^2) (Log[a-I (Sqrt[-(a^2)+(c^2)])]-Log[a+I (Sqrt[-(a^2)+(c^2)])])))/(Sqrt[-(a^2)+(c^2)])),0<a<c]

With[{a=1,c=3},((a π (2 (Sqrt[-(a^4)+(a^2) (c^2)])+I (c^2) (Log[a-I (Sqrt[-(a^2)+(c^2)])]-Log[a+I (Sqrt[-(a^2)+(c^2)])])))/(Sqrt[-(a^2)+(c^2)]))]//N

ProlateSpheroidSurfaceArea2[a_,c_]:=((a π (2 (Sqrt[-(a^4)+(a^2) (c^2)])+I (c^2) (Log[a-I (Sqrt[-(a^2)+(c^2)])]-Log[a+I (Sqrt[-(a^2)+(c^2)])])))/(Sqrt[-(a^2)+(c^2)]))

ProlateSpheroidSurfaceArea2[1,3]//N