Flat point

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A point on regular surface at which the osculating paraboloid degenerates to a plane. At a flat point the Dupin indicatrix is not defined, the Gaussian curvature is equal to zero and the second fundamental form and all remaining curvatures are also equal to zero.

A point at which the torsion of a curve vanishes is called a flat point of the spatial curve.


A flat point is sometimes called a planar point. The terminology of course derives from the curvature zero property.


[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
How to Cite This Entry:
Flat point. D.D. Sokolov (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098