SingularLineInAPseudoSphere::usage="Find the singular line (recoil edge) in the pseudo sphere: Sin[u]Cos[v],Sin[u]Sin[v],Cos[u]+LogTanReferences:Pogorélov,A.V.GeometríaDiferencial.EditorialMir:p.105,Moscú,1977;Personal";
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Condition to obtain the singular points: [u,v][u,v] == 0.
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r[u_,v_]:=Sin[u]Cos[v],Sin[u]Sin[v],Cos[u]+LogTan;[u_,v_]=D[r[u,v],u];[u_,v_]=D[r[u,v],v];Solve[[u,v][u,v]==0,{u,v}]
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u-,u-,u,u
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The answer is u = , because if it is replace in r[, v] results {Cos[v], Sin[v], 0} which is a curve that divides to lthe pseudo sphere in two leafs, there are the singular points.
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Animation:
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ManipulateShowRevolutionPlot3DSin[u],0,LogTan+Cos[u],{u,0,2π},{θ,0,θ1},RevolutionAxis{0,0,1},PlotStyle->Opacity[0.6],MeshFalse,BoundaryStyleDirective[Black,Thick],PerformanceGoal"Quality",ColorFunction"BlueGreenYellow",ParametricPlot3D[{Cos[v],Sin[v],0},{v,0,θ1},PlotStyle->{Red,Thick}],ParametricPlot3D[t{0,0,1},{t,-5,5},ColorFunction->"BlueGreenYellow"],Graphics3D[{Black,Ball[{0,0,0},0.05],Ball[{Cos[θ1],Sin[θ1],0},0.05],Dashed,Thick,Line[{{0,0,0},{Cos[θ1],Sin[θ1],0}}]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange1,ViewPoint{1.3,-1,1.},Style["Recoil edge (in red):",Bold,Large],{{θ1,0.01,"Value (θ)"},0.001,2π,0.01},ControlPlacementTop
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