hyperbola

It's an hyperbola, but not by that much...

It works- but if it is not at the focal point it might not work, so my suggestion is drawing the hyperbola first and then finding the focal point.

A "small" section of a hyperbola is very similar to a small section of a circle that is very similar to a small section of a parabola.

It'll approximate a parabola if you throw at exactly the escape velocity, and a hyperbola if you throw even harder than that.

Picture the event horizon as a hyperbola, and the test particle as an asymptote: it can never intersect it.> a test of the laws of physics can distinguish X from an inertial frame in an idealized, gravity-free universeWell duh. There are no inertial frames that cross the horizon.

Well, I have a clever way to explain the relationship between areas under a hyperbola and continuously compounded interest:dy/dx = 1/xdx/dy = xHopefully that isn't too confusing.

An observer at constant r close to the horizon has to accelerate very hard to stay at the same radius; that means a line of constant r in Frame X is going to look like a hyperbola in a spacetime diagram.

That is the key feature that the article gets wrong.> Then law J shows that a locally inertial frame relative to which the particle is at rest can’t even in principle extend below the horizonThe horizon forms a hyperbola in any such inertial frame.