Also, as an aside to the tangent, tangent is a strange phrase, since it doesn't actually touch the main point. Should be polar line or somesuch.

"Tangent" is perfectly appropriate -- it touches a point somewhere on the curve of the main argument, and then diverges. There is something that made the association with the tangent.

And, to further overextend this metaphor, this implies that if someone's argument is rough enough (i.e. not differentiable), then it's not even possible to go off it on a tangent.