Umbilical point

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An elliptic point of a surface at which the osculating paraboloid degenerates into a paraboloid of revolution. At an umbilical point the normal curvatures in all directions are equal, and the Dupin indicatrix is a circle. An umbilical point is sometimes called a spherical point or circular point.


At flat points (cf. Flat point) the osculating paraboloid degenerates to a plane. Often flat points are also called umbilical. However, the Dupin indicatrix is not defined at flat points.


[a1] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
How to Cite This Entry:
Umbilical point. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098