Umbilic
In the differential geometry of surfaces in three dimensions, umbilics or umbilical points are points on a surface that are locally spherical. At such points the normal curvatures in all directions are equal, hence, both principal curvatures are equal, and every tangent vector is a ''principal direction''. The name "umbilic" comes from the Latin ''umbilicus'' (navel). Umbilic points generally occur as isolated points in the elliptical region of the surface; that is, where the Gaussian curvature is positive. The sphere is the only surface with non-zero curvature where every point is umbilic. A flat umbilic is an umbilic with zero Gaussian curvature. The monkey saddle is an example of a surface with a flat umbilic and on the plane every point is a flat umbilic. A closed surface topologically equivalent to a torus may or may not have zero umbilics, but every closed surface of nonzero Euler characteristic, embedded smoothly into Euclidean space, has at least one umbilic. A famous co ...
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