AffGamma[β_,ν_]:=
{β,ν,Log[Gamma[ν]]-ν Log[β]}

S=ParametricPlot3D[AffGamma[β,ν]//Evaluate,
{β,0.00001,4},{ν,0.0001,3}, ViewPoint->{3,2,1}, BoxRatios->{1,1,1},AxesLabel->{ν,κ},Mesh->None,LabelStyle->Directive[Medium,Bold],FaceGrids->All]

R=ParametricPlot3D[{β-((0.3` Cos[theta])/((Sqrt[1+(1/(β^2))]) (Sqrt[(1/(β^4))]) (β^3))),1-((0.3` Sin[theta])/((Sqrt[(1/(β^4))]) (β^2))),-((0.3` Cos[theta])/((Sqrt[1+(1/(β^2))]) (Sqrt[(1/(β^4))]) (β^2)))-Log[β]}
//Evaluate,
{β,0.0001,4},{theta,0,2Pi},AxesLabel->{ν,κ},Mesh->None,LabelStyle->Directive[Medium,Bold],FaceGrids->All,
PlotPoints->{20,20},ViewPoint->{3,2,1}, BoxRatios->{1,1,1}]

Show[S,R]