Imagine a cone (of infinite height) with the vertex in the origin of a 3d system. The cone is cut in 2 equal parts by the XY plane.
[Points of the XY plane, can be interpreted as complex numbers, or just as pairs (X, Y), whatever is more convenient.]
Now my question is: does exist a 3d curve starting from the origin (cone vertex) and wrapping around the cone (say, in a way possibly similar to a sort of "slanted" conic spiral) such that, if we consider the projection on the plane XY of each Z-positive "half spiral", it comes out to be a hyperbola symmetric with respect to the y = x line, at least in the first quadrant of the XY plane?
I am wondering if such a curve exists, has a name, and, if yes, what is a general form it could take (possibly as parametric representation, or complex function, whatever is more convenient or elegant to use)?
PS. I am looking for a unique continuous curve going around the cone (similar to a conic spiral), say something like: (x(theta), y(theta), z(theta) ) giving rise to all these symmetric hyperbolic projections on XY first quadrant. Imagine a point that starts from the origin and moves in some sort of spiral towards infinity, in such a way to project those symmetric hyperbolas on plane XY (wrt y=x), at least when it traverses the "positive half" of each loop (it can use the second, "negative half", to "advance", in whatever way).