Oblate Spheroid Geodesic

Author

Eric W. Weisstein
May 3, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/DifferentialGeometry/OblateSpheroidGeodesic.nb.

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

©2005 Wolfram Research, Inc. except for portions noted otherwise

Geodesic Equations

x[u_,v_]:=a Sin[v]Cos[u]
y[u_,v_]:=a Sin[v]Sin[u]
z[u_,v_]:=c Cos[v]

P[vars_List,{u_,v_}]:=Plus@@(D[#[u,v],u]^2&/@vars)
Q[vars_List,{u_,v_}]:=Plus@@(D[#[u,v],u]D[#[u,v],v]&/@vars)
R[vars_List,{u_,v_}]:=Plus@@(D[#[u,v],v]^2&/@vars)

FullSimplify[#[{x,y,z},{u,v}]&/@{P,Q,R}/. c^2->a^2(1- e^2)]

Simplify[(1/2) (a^2) (2-(e^2)+(e^2) Cos[2 v])-a^2(1-e^2 Sin[v]^2)]

i=FullSimplify[Sqrt[R[{x,y,z},{u,v}]/(P[{x,y,z},{u,v}]^2-(Subscript[c, 1])^2 P[{x,y,z},{u,v}])]/. c^2->a^2(1- e^2)]

Simplify[(i-Sqrt[1/(Subscript[c, 1])^2(1-e^2 Sin[v]^2)/((a/(Subscript[c, 1]))^2 Sin[v]^2-1)]/Sin[v])]

Simplify[(Sqrt[((-(e^2)+((Csc[v])^2))/((a^2) ((Sin[v])^2)-(Subscript[c, 1]^2)))])- (Sqrt[Csc[v]^2((1-(e^2) ((Sin[v])^2))/((a^2) ((Sin[v])^2)-(Subscript[c, 1]^2)))])]

Integral

∫((Sqrt[((1-(e^2) (sin^2)(v))/((d^2) (sin^2)(v)-1))])/(sin(v)))\[DifferentialD]v

Integrate[Sqrt[(1-e^2 Sin[v]^2)/(d^2 Sin[v]^2-1)]/Sin[v],v]

FullSimplify[%]

PowerExpand[%]

FullSimplify[%]

Together//@%