an umbilic point
/ UM-bil-ik /
At most points of a surface there is a clear 'most-curved' direction and a perpendicular 'least-curved' one — a saddle clearly bends up one way and down the other. But at some special points the surface curves by the SAME amount in every direction, with no preferred axis at all, like the very top of a perfectly round dome. Such a point is an umbilic point. The name comes from the Latin for navel — the spot where the surface is locally as round as a sphere.
Precisely, a point is umbilic when the two principal curvatures are equal, k_1 = k_2. Then the normal curvature k_n(theta) = k_1 cos^2 theta + k_2 sin^2 theta is the same constant for every direction theta, so every direction is a principal direction and there is no distinguished one. Equivalently, the second fundamental form is a scalar multiple of the first (II = k * I) at that point. There are two flavours. If k_1 = k_2 is nonzero, the surface looks locally spherical (a spherical umbilic). If k_1 = k_2 = 0, it is a flat umbilic (also called a planar point), where the surface is flat to second order, like the centre of the monkey saddle z = x^3 - 3xy^2.
Umbilics are where the structure of lines of curvature breaks down — since every direction is principal, the neat perpendicular grid of curvature lines has no way to choose a direction and instead swirls into characteristic singular patterns around the umbilic. A clean global theorem: a connected surface ALL of whose points are umbilic is necessarily part of a plane (all flat umbilics) or part of a sphere (all spherical umbilics) — these are the only totally umbilic surfaces. On a general ellipsoid there are exactly four isolated umbilic points. One caution: 'umbilic' is about the principal curvatures being equal, NOT about the point being special in any topological or singular sense — the surface is perfectly smooth and regular there; what fails is only the labelling of principal directions.
Every single point of a sphere is umbilic: the sphere curves by 1/R in all directions, so k_1 = k_2 = 1/R everywhere — that is why a sphere has no distinguished lines of curvature. By contrast, a general ellipsoid (with three different axis lengths) is umbilic at only four isolated points; everywhere else it has two genuinely different principal curvatures. The four umbilics of an ellipsoid are the points where its lines of curvature spiral into characteristic patterns.
Every point of a sphere is umbilic; a general ellipsoid has exactly four.
An umbilic point is perfectly smooth and regular — nothing is 'wrong' with the surface there. What becomes undefined is only the pair of principal DIRECTIONS, because when k_1 = k_2 every direction bends equally and none is singled out as 'most curved.'