Geodesic

Author

Eric W. Weisstein
May 3, 2004

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

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

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

Code

Geodesic[{x_,y_,z_},{u_,v_}]:=Module[{
P=Simplify[D[x,u]^2+D[y,u]^2+D[z,u]^2],Q=Simplify[D[x,u]D[x,v]+D[y,u]D[y,v]+D[z,u]D[z,v]],R=Simplify[D[x,v]^2+D[y,v]^2+D[z,v]^2]},
{P,Q,R}]

GeodesicPlot[{x_,y_,z_},{u_,umin_,umax_},{v_,vmin_}]:=Module[{P,Q,R},
{P,Q,R}=Geodesic[{x,y,z},{u,v}];
NDSolve[{(D[P,v]+2v'[u]D[Q,v]+v'[u]^2D[R,v])/2Sqrt[P+2Q v'[u]+Rv'[u]^2]-D[(Q+Rv'[u])/Sqrt[P+2Qv'[u]+Rv'[u]^2],u]==0,
u[vmin]==umin
},v[u],{u,umin,umax}]
]

GeodesicPlot[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{u,0,1},{v,0}]

?NDSolve

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

Sphere

{P,Q,R}=Geodesic[a{Cos[u]Sin[v],Sin[u]Sin[v],Cos[v]},{u,v}]

PowerExpand[Sqrt[R/(P^2-(Subscript[c, 1])^2 P)]]

PowerExpand[Sqrt[Simplify[%]]]

FullSimplify[Subtract@@{-ArcTan[Cos[v]/Sqrt[(Subscript[a, 1])^2 Sin[v]^2-((Subscript[c, 1])^2)]],-ArcSin[Cot[v]/Sqrt[(a/(Subscript[c, 1]))^2-1]]}]

Plot[{-ArcTan[Cos[x]/Sqrt[4 Sin[x]^2-1]],-ArcSin[Cot[x]/Sqrt[4-1]]},{x,.5,1.5},PlotStyle->{{},{Thickness[0.01],Dashing[{.03,.03}]}}]

u[v_]:=-ArcSin[Cot[v]/Sqrt[(a/(Subscript[c, 1]))^2-1]]+(Subscript[c, 2])

Simplify[P u'[v]^2+2 Q u'[v]+R]

Some Surface

{P,Q,R}=Geodesic[{a Cos[u]Sin[v],a Sin[u]Sin[v],c Cos[v]},{u,v}]

Some Other Surface

{P,Q,R}=Geodesic[{a Cos[u]Sin[v],b Sin[u]Sin[v],c Cos[v]},{u,v}]