SterProj[{z_,w_}]:=Module[{x=Re[z],y=Im[z],u=Re[w],v=Im[w]},{(x/(1-v)),(y/(1-v)),(u/(1-v))}//ComplexExpand]
σ[{z_,w_}]:={w,z}

z=Exp[I α]Sin[γ];
w=Exp[I β]Cos[γ];

 g[Α_]=SterProj@({z,w})/.α->Α;
Show[ParametricPlot3D[g[0],{γ,0,π/4},{β,0,2π},PlotRange->{{-π,π},{-π,π},{-π,π}}],ParametricPlot3D[g[π],{γ,0,π/4},{β,0,2π},PlotStyle->Blue,PlotRange->{{-π,π},{-π,π},{-π,π}}]]

SterProj[{z,w}]

f[Γ_]=SterProj/@{{z,w},σ@{z,w}}/.γ->Γ;
ParametricPlot3D[f[π/4],{α,0,2π},{β,0,2π}]

{g[0],g[π]}

h[Β_]=SterProj@({z,w})/.β->Β;
Show[Table[ParametricPlot3D[h[2π n/10],{γ,0,π/4},{α,0,2π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{n,1,10}]]

g[Α_]=SterProj@{z,w}/.α->Α;
Show[Table[ParametricPlot3D[g[2π n/4],{γ,0,π/4},{β,0,2π},PlotRange->{{-π,π},{-π,π},{-π,π}}],{n,1,4}]]