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Asymptotic direction

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A direction on a regular surface in which the curvature of the normal section of the surface is zero. For the direction $du\colon dv$ at a point $P$ to be an asymptotic direction, the following condition is necessary and sufficient:

$$Ldu^2+2Mdudv+Ndv^2=0,$$

where $u$ and $v$ are interior coordinates on the surface, while $L,M$ and $N$ are the coefficients of the second fundamental form of the surface, calculated at $P$. At an elliptic point of the surface the asymptotic directions are imaginary, at a hyperbolic point there are two real asymptotic directions, at a parabolic point there is one real asymptotic direction, and at a flat point any direction is asymptotic. Asymptotic directions are self-conjugate directions (cf. Conjugate directions).

References

[1] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)


Comments

References

[a1] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981)
[a2] D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950)
How to Cite This Entry:
Asymptotic direction. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Asymptotic_direction&oldid=32640
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article