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Equation: , i.e.  (see the notations).

An umbilic of a surface is a point where the curvature radii of the normal sections are all equal (in other words, it is either an elliptic point where the Dupin indicatrix is circular, or a planar point).
 - the intersection points between a surface of revolution and the axis of revolution.
 - all the points of the sphere (conversely, the surfaces all the points of which are umbilics are portions of spheres or planes).

See also the umbilics of the ellipsoid, of the elliptic paraboloid, of the one-sheeted or two-sheeted hyperboloid.

cf. A. Gullstrand, Zur Kenntniss der Kreispünkte, Acta Mathematica, 1905, p. 59 à 100.
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© Robert FERRÉOL 2017