Intersection of y-x^2 and x=0 in projective space

p=ContourPlot3D[y z-x^2==0,{x,-2,2},{y,-2,2},{z,-2,2},AxesLabel->{"x","y","z"},ViewPoint->{2.5,1,1},Mesh->None]

l=ParametricPlot3D[{0,y,z},{y,-2,2},{z,-2,2},Mesh->None];
point=ParametricPlot3D[{0,y,0},{y,-2.5,2.5}];

Show[p,l,point,PlotRange->{{-2.5,2.5},{-2.5,2.5},{-2.5,2.5}}]

The twisted cubic.

ParametricPlot3D[{t,t^2,t^3},{t,-1,1},PlotStyle->Thick,BoxRatios->{1,1,1}]

The three standard slices of the cone over the parabola, y-x^2.

p=ContourPlot3D[y z-x^2==0,{x,-2,2},{y,-2,2},{z,-2,2},AxesLabel->{"x","y","z"},ViewPoint->{2.5,1,1},Mesh->None];
ax=ParametricPlot3D[{1,y,z},{y,-2,2},{z,-2,2},PlotRange->2,PlotStyle->Opacity[0.8],Mesh->None];
ay=ParametricPlot3D[{x,1,z},{x,-2,2},{z,-2,2},PlotRange->2,PlotStyle->Opacity[0.8],Mesh->None];
az=ParametricPlot3D[{x,y,1},{x,-2,2},{y,-2,2},PlotRange->2,PlotStyle->Opacity[0.8],Mesh->None];

Show[p,az]

Show[p,ay]

Show[p,ax]

The three standard slices of the cone over the hyperbola xy-1.

h=ContourPlot3D[x y-z^2==0,{x,-3,3},{y,-3,3},{z,-3,3},AxesLabel->{"x","y","z"},Mesh->None,RegionFunction->Function[{x,y},Abs[x+y]≤3]];
bx=ParametricPlot3D[{1,y,z},{y,-3,3},{z,-3,3},PlotRange->3,PlotStyle->Opacity[0.8],Mesh->None];
by=ParametricPlot3D[{x,1,z},{x,-3,3},{z,-3,3},PlotRange->3,PlotStyle->Opacity[0.8],Mesh->None];
bz=ParametricPlot3D[{x,y,1},{x,-3,3},{y,-3,3},PlotRange->3,PlotStyle->Opacity[0.8],Mesh->None];

Show[h,bz,ViewPoint->{3,1,1}]

Show[h,bx,ViewPoint->{3,1,1}]

Show[h,by,ViewPoint->{3,1,1}]

Trying to see points at infinity on the circle, x^2+y^2=1.  They are (+/- i, 1, 0).

c=ContourPlot3D[x^2+y^2-z^2==0,{x,-2,2},{y,-2,2},{z,-2,2},AxesLabel->{"x","y","z"},ViewPoint->{2.5,1,1},Mesh->None];
Show[c,ay]