It's not really necessary to go into it to that degree. I'll grant that Antonsen is simply using a simple diagram to make a point, and the ersatz-hyperbola is close enough for a PowerPoint show. There's no need to rub more salt into the wound. The reason I mentioned it now is because Arfa Brane was beginning to ask about actual hyperbolic components, such as asymptotes. That geometric shape has no asymptotes, and it is not a continuous function. In fact, it's not a function at all. A hyperbola is a continuous object, extending to infinity. That diagram cannot. It has discrete values (No, you can't simply add more lines to it. I could go into this in further detail if you have doubts). Now, Antonsen never talked about those things, so I can't hang that on him, which is why it wasn't necessary to get pedantic about it. But calling it a parabola - as opposed to a hyperbola - that is a pure right/wrong error.