Asymptotic direction
A direction on a regular surface in which the curvature of the normal section of the surface is zero. For the direction at a point
to be an asymptotic direction, the following condition is necessary and sufficient:
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where and
are interior coordinates on the surface, while
and
are the coefficients of the second fundamental form of the surface, calculated at
. 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) |
Asymptotic direction. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_direction&oldid=32640