Bill: I gather from the recent addition of mouth shape as an alias that that term is used by Bozulich and van Zeijst for this shape. Does parabola also exist in the published literature? If not, maybe parabola should be the alias.
Tirian: Off the top of my head, I recall that Davies called Black 1 the eye-making tesuji.
Anyway, the mathematician in me is recoiling at "parabola". It's far too granular to have a good fight over it, but the curve of the black stones seems more hyperbolic or logarithmic to me. Indeed, it may be that the shape is be one branch of a hyperbola in the "taxicab geometry" and the key point its associated focal point.
Velobici: I would be happy to move the content (with light edits) from parabola to mouth shape and to fix the links on other pages. Didnt do so because I do not remember seeing either term used much and so dont know which should be the alias and which the content page. Any Chinese, Japanese, or Korean (CJK) names for this shape?
ilan: Well, catenary might be better than parabola, since you can actually observe those in nature. Anyway, if you did call it a parabola, what would be its latus rectum?