Can the two branches of a single hyperbola hypothetically intersect

Question

I know it is a loaded question and there has to be some ignoring of basic rules such as the fact that it would no longer be classified as a "conic section", but I am wondering if there is any mathematical possibility that the two branches of a hyperbola can theoretically intersect. My first thought would have been that it would have to utilize a concept such as imaginary numbers however as they are graphed similarly to real numbers I am not sure how it would work. My other idea that would make the idea similar to a conic section is that if one were to drop the tip of one of the cones that make up the intersection point inside of the other cone and cut the cones perpendicularly, that would produce a 'hyperbola' in which the branches intersect. If it is even mathematically possible, what is the math behind it?

Answer 1

The two branches of course don’t intersect in the Euclidean plane $\mathbb R^2$, but we can extend $\mathbb R^2$ to the projective plane $\mathbb P^2$ by adding some points and a line: for each family of parallel lines in $\mathbb R^2$ add a “point at infinity” that lies on all of those lines, and add a “line at infinity” on which all of these additional points, and no others, lie. A hyperbola will then include a pair of points at infinity that lie on its asymptotes. Not only do the two branches of a hyperbola intersect in $\mathbb P^2$, but it is a single connected curve.

The projective plane “wraps around” the way some old space battle video games did. If you move along a line, eventually you reach the point at infinity, and if you keep going from there, you continue along the line “from the other side.” With this in mind, just as in Euclidean geometry an ellipse can be viewed as a stretched-out circle, in projective geometry a parabola can be seen as a circle that’s been stretched out to the line at infinity, and a hyperbola as a circle that’s been stretched “through” the line at infinity. Conics can thus be distiguished by their intersection with the line at infinity: hyperbolas have two, parabolas one and ellipses none.

On the other hand, the line at infinity is a first-class citizen: there’s nothing about it that’s really different from any of the “finite” lines. Indeed, the choice of a line at infinity in the projective plane is arbitrary. With different choices, the same curve might be an ellipse, parabola or hyperbola. From the point of view of projective geometry, then, there is only one kind of nondegenerate conic.