Why the name "umbilic"?

Question

Umbilic points are points on a surface at which the principle curvatures of the surface are equal. "Umbilic(al)" refers to the navel/belly button. But why do we call these points so? What about the surface is reminiscent of a belly button at these points? I can only think of a swollen tummy, which might be round around the belly button, much as some surfaces are locally spherical around an umbilic point.

Answer 1

At those points, the surface resembles either an "innie" or an "outie".

Answer 2

Another meaning of "umbilical" is "extremely close; inseparable." If you think of a surface as embedded in $\mathbb{R}^3$, at umbilic points, the surface is intimately attached to a bounding sphere.

Answer 3

The word "ombilic" is used already in Gaston Darboux's text from the 19th century. See this link.

Umbilic point is an entry in 1700 in Joseph Moxon’s dictionary of mathematics: "Umbilique Points, or the 2 Focus or Centre-Points in an Elipsis." (see http://jeff560.tripod.com/f.html).