The real projective plane is a sphere with antipodal points identified.

sphere=ParametricPlot3D[{Cos[s]Sin[t],Sin[s]Sin[t],Cos[t]},{s,0,2Pi},{t,0,Pi},Mesh->None,PlotStyle->Opacity[0.5]];

r=1.2; plane=ParametricPlot3D[{x,y,1},{x,-r,r},{y,-r,r},PlotStyle->Opacity[0.3]];

parabola=ParametricPlot3D[{x,x^2,1},{x,-1.2,1.2},PlotStyle->{Thick,Red}];

projparab=ParametricPlot3D[{{x/Sqrt[1+x^2+x^4],x^2/Sqrt[1+x^2+x^4],1/Sqrt[1+x^2+x^4]},{-x/Sqrt[1+x^2+x^4],-x^2/Sqrt[1+x^2+x^4],-1/Sqrt[1+x^2+x^4]}},{x,-3,3},PlotStyle->{{Thick,Red},{Thick,Red}}];

w=0.8;line=ParametricPlot3D[{s w,s w^2,s},{s,-1.5,1.5},PlotStyle->{Thick,Blue}];

Show[sphere,plane,parabola,projparab,line,Background->Black,Ticks->None,Boxed->False,PlotRange->1.3]

ht=0.5;hyperbola=ParametricPlot3D[{t,ht/t,1},{t,-2,2},RegionFunction->Function[{t},t>0.3||t<-0.3],PlotStyle->{Thick,Red}];

projhyperb=ParametricPlot3D[{{t/Sqrt[1+t^2+ht^2/t^2],(ht/t)/Sqrt[1+t^2+ht^2/t^2],1/Sqrt[1+t^2+ht^2/t^2]},{-t/Sqrt[1+t^2+ht^2/t^2],-(ht/t)/Sqrt[1+t^2+ht^2/t^2],-1/Sqrt[1+t^2+ht^2/t^2]}},{t,-3,3},RegionFunction->Function[{t},t>0.2||t<-0.2],PlotStyle->{{Thick,Red},{Thick,Red}}];

Show[sphere,plane,hyperbola,projhyperb,line,Background->Black,Ticks->None,Boxed->False,PlotRange->1.5]