0<a<π/4 vagy π/4<π/2

Vr[a_,s_]:=ArcSinh[((Cos[a])/(Sqrt[Abs[Cos[2a]]]))Sinh[s(Sqrt[Abs[Cos[2a]]])]]/;0<a<(π/4)∨(π/4)<a<(π/2)
Vϑ[a_,s_]:=-ArcTan[((Sin[a])/(Sqrt[Abs[Cos[2a]]]))Tan[s(Sqrt[Abs[Cos[2a]]])]]/;0<a<(π/4)∨(π/4)<a<(π/2)
Vφ[a_,s_]:=s 2Sin[a]+Vϑ[a,s]/;0<a<(π/4)∨(π/4)<a<(π/2)
Vx[a_,s_]:=Tan[Vφ[a,s]]/;0<a<(π/4)∨(π/4)<a<(π/2)
Vy[a_,s_]:=Tanh[Vr[a,s]]((Cos[Vϑ[a,s]-Vφ[a,s]])/(Cos[Vφ[a,s]]))/;0<a<(π/4)∨(π/4)<a<(π/2)
Vz[a_,s_]:=Tanh[Vr[a,s]]((Sin[Vϑ[a,s]-Vφ[a,s]])/(Cos[Vφ[a,s]]))/;0<a<(π/4)∨(π/4)<a<(π/2)

a=π/4

Különböző számú változóval lehet, sőt célszerű  definiálni ugyanazt a fvt, a kisbetűs matban meg nem.

Limit[ArcSinh[((Cos[a])/(Sqrt[Abs[Cos[2a]]]))Sinh[s(Sqrt[Abs[Cos[2a]]])]],a->π/4]

Ennek felhasználásával taln még kevesebbet kell írni.

Vr[s_]:=ArcSinh[(s/(Sqrt[2]))]
Vϑ[s_]:=-ArcTan[(s/(Sqrt[2]))]
Vφ[s_]:=s (Sqrt[2])-ArcTan[(s/(Sqrt[2]))]
Vx[s_]:=Tan[Vφ[s]]
Vy[s_]:=Tanh[Vr[s]]((Cos[Vϑ[s]-Vφ[s]])/(Cos[Vφ[s]]))
Vz[s_]:=Tanh[Vr[s]]((Sin[Vϑ[s]-Vφ[s]])/(Cos[Vφ[s]]))

(*s:=1.5;nem adunk neki globálisan értéket$*)
t1:=ParametricPlot3D[{{(Sqrt[((Vy[a, s])^2)+((Vz[a, s])^2)])Cos[ω],(Sqrt[((Vy[a, s])^2)+((Vz[a, s])^2)])Sin[ω],Vx[a,s]}, {(Sqrt[((Vy[a, s])^2)+((Vz[a, s])^2)])Cos[ω],(Sqrt[((Vy[a, s])^2)+((Vz[a, s])^2)])Sin[ω],-Vx[a,s]}}/.{s->1.5},{a, π/4, π/2}, {ω, -π, π }, PlotStyle->{{LightBlue,Specularity[White,40], Opacity[0.7]},{ LightBlue,Specularity[White,40], Opacity[0.7]}}, Mesh->None]

t1

t2:=ParametricPlot3D[{{(Sqrt[((Vy[a, s])^2)+((Vz[a,s])^2)])Cos[ω],(Sqrt[((Vy[a,s])^2)+((Vz[a,s])^2)])Sin[ω],Vx[a,s]}, {(Sqrt[((Vy[a,s])^2)+((Vz[a,s])^2)])Cos[ω],(Sqrt[((Vy[a,s])^2)+((Vz[a,s])^2)])Sin[ω],-Vx[a,s]}}/.{s->1.5},{a, 0, π/4}, {ω, -π, π },
PlotStyle->{{Orange,Specularity[White,10], Opacity[0.7]},{ Orange,Specularity[White,10], Opacity[0.7]}}, Mesh->None]

t2

t3:=ParametricPlot3D[{{(Sqrt[((Vy[ s])^2)+((Vz[s])^2)])Cos[ω],(Sqrt[((Vy[s])^2)+((Vz[s])^2)])Sin[ω],Vx[s]}, {(Sqrt[((Vy[s])^2)+((Vz[s])^2)])Cos[ω],(Sqrt[((Vy[s])^2)+((Vz[s])^2)])Sin[ω],-Vx[s]}},{s,0,π/2}, {ω, -π, π }, PlotStyle->{{Yellow,Specularity[White,10], Opacity[0.6]},{Yellow,Specularity[White,10], Opacity[0.6]}}, Mesh->None]

t3

Show[t1,t2,t3]