# Difference between revisions of "Parabolic point"

From Encyclopedia of Mathematics

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A point on a regular surface at which the [[Osculating paraboloid|osculating paraboloid]] degenerates into a [[Parabolic cylinder|parabolic cylinder]]. At a parabolic point the [[Dupin indicatrix|Dupin indicatrix]] is a pair of parallel straight lines, the [[Gaussian curvature|Gaussian curvature]] is equal to zero, one of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) vanishes, and the coefficients of the [[Second fundamental form|second fundamental form]] satisfy the equation | A point on a regular surface at which the [[Osculating paraboloid|osculating paraboloid]] degenerates into a [[Parabolic cylinder|parabolic cylinder]]. At a parabolic point the [[Dupin indicatrix|Dupin indicatrix]] is a pair of parallel straight lines, the [[Gaussian curvature|Gaussian curvature]] is equal to zero, one of the principal curvatures (cf. [[Principal curvature|Principal curvature]]) vanishes, and the coefficients of the [[Second fundamental form|second fundamental form]] satisfy the equation | ||

− | + | $$LN-M^2=0.$$ | |

## Latest revision as of 19:11, 12 April 2014

A point on a regular surface at which the osculating paraboloid degenerates into a parabolic cylinder. At a parabolic point the Dupin indicatrix is a pair of parallel straight lines, the Gaussian curvature is equal to zero, one of the principal curvatures (cf. Principal curvature) vanishes, and the coefficients of the second fundamental form satisfy the equation

$$LN-M^2=0.$$

#### Comments

#### References

[a1] | W. Klingenberg, "A course in differential geometry" , Springer (1978) pp. 50–51 (Translated from German) |

[a2] | R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. 132 |

**How to Cite This Entry:**

Parabolic point.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Parabolic_point&oldid=18070

This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article