I'm sure we've all seen the image below that illustrates the creation of the four conic sections. Although I've seen this multiple times throughout my education, I find it odd that the following case has never been discussed: If we take the vertical plane that forms a hyperbola and tilt it just a few degrees in a way that still crosses both sides of the cone, shouldn't we get an asymmetric hyperbola? Being an astronautical engineer, I've studied how (symmetric) hyperbolas model highly eccentric orbits of celestial objects. What physical processes can be modelled by asymmetric hyperbolas (assuming they exist)?
Best Answer
The comments by the OP indicate that he may be confused about the nature of the branches of a hyperbola. A branch of a hyperbola is never a parabola. One way of seeing this is to notice that a branch of a hyperbola has a pair of transverse asymptotic lines, whereas a parabola does not have asymptotic lines at all.
When one tilts the vertical plane it may look as if one is creating an asymmetric hyperbola, but this is an illusion. The illusion is possibly due to the fact that one tends to think of the center of the hyperbola is being "at the same level" as the vertex of the cone. What actually happens is that the center of the hyperbola travels away from this level as the plane is tilted more and more.