In the projective plane sense, a (projective) hyperbola is just an elliptic cone in 3D with the projection point as its cone peak $O$, and with the "level plane" (onto which to project) tilted so that it intersects with both halves of the cone, where the projection image on the level plane is a hyperbola curve in the usual sense, as the following image illustrates:
Now I want to show that both the asymptotes of the hyperbola, in the projective plane, intersect the hyperbola (the cone) exactly once, ie they are both tangent to the hyperbola (the cone) at infinity.
I have shown that the cone has exactly two infinity points, both of which has exactly a projective tangent line (a tangent plane to the cone). Where I have trouble is I don't know why these two tangent lines (planes) will exactly project onto the asymptotes on the level plane. Sure, one thing to note that is they both intersect the hyperbola (the cone) exactly once at infinity and nowhere on the "ordinary plane" (the level plane), just as both the asymptotes don't intersect the hyperbola curve on the level plane. The problem is that there are unfortunately many other lines other than the asymptotes that don't intersect the hyperbola on the level plane, and it seems they can't be distinguished in this way.
edit just worked out a perturbation technique that can tell apart asymptotes and other lines that don't intersect the hyperbola, observing that asymptotes, if just perturbed a little bit, will intersect the hyperbola curve, whereas other lines are "stable" under perturbation. So this problem is basically solved, but new approaches are welcome and appreciated too.