I like your suggestion regarding asymptotic JH, but have never encountered it.

Hyperbole and parable have the same roots as hyperbola and parabola and give us hyperbolic and parabolic to describe either the rhetorical or the mathematical use. So why not use asymptote and asymptotic similarly? I guess tangent and tangential are the most widely used examples of this linking of curves with language. Are there others? How about a hyperparable for an allegory that really takes things beyond credibility, matched with hyperbolicparaboloid?