Difference between revisions of "Normal section"
From Encyclopedia of Mathematics
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| − | ''of a smooth surface | + | ''of a smooth surface $\Phi$ at a point $P$ in a direction $\ell$'' |
| − | The section of | + | 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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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Klingenberg, "A course in differential geometry" , Springer (1978) (Translated from German)</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> B.-Y. Chen, "Geometry of submanifolds" , M. Dekker (1973)</TD></TR> | ||
| + | </table> | ||
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| + | {{TEX|done}} | ||
Revision as of 22:14, 15 December 2016
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.
Comments
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=40020
Normal section. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normal_section&oldid=40020
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article