Difference between revisions of "Normal section"
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The section of $\Phi$ by the plane passing through the normal to the surface (cf. [[Normal space (to a surface)]]) at $P$ and through the direction $\ell$ in the [[tangent plane]] to $\Phi$ at $P$. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see [[Curvature]]; [[Normal curvature]]). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension. | The section of $\Phi$ by the plane passing through the normal to the surface (cf. [[Normal space (to a surface)]]) at $P$ and through the direction $\ell$ in the [[tangent plane]] to $\Phi$ at $P$. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see [[Curvature]]; [[Normal curvature]]). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension. | ||
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Latest revision as of 07:59, 16 April 2023
of a smooth surface $\Phi$ at a point $P$ in a direction $\ell$
The section of $\Phi$ by the plane passing through the normal to the surface (cf. Normal space (to a surface)) at $P$ and through the direction $\ell$ in the tangent plane to $\Phi$ at $P$. The task of studying the local structure of a surface can be reduced to the same task for the family of curves formed by the normal sections of the surface at a given point in various directions (see Curvature; Normal curvature). The method of studying the local structure by means of normal sections can be generalized to surfaces of arbitrary dimension and arbitrary codimension.
References
| [a1] | W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German) |
| [a2] | B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973) |
How to Cite This Entry:
Normal section. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_section&oldid=53817
Normal section. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_section&oldid=53817
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article