# asymptotic directions

## asymptotic directions

[‚ā·sim′täd·ik də′rek·shənz]
(mathematics)
For a hyperbolic point on a surface, the two directions in which the normal curvature vanishes; equivalently, the directions of the asymptotic curves passing through the point.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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At last, we note here that the global exponential stability of traveling curved fronts in the sense of Theorem 3isa difficult problem, since the level set of the traveling curved fronts [PHI](z, y) of (1) have two asymptotic directions as [absolute value of (z)] [right arrow] +[infinity], and both directions make an angle with the negative y-axis, which is different from the case of planar traveling fronts (see ).
In the differential geometry of surfaces, an asymptotic curve is formally defined as a curve on a regular surface such that the normal curvature is zero in the asymptotic direction. Asymptotic directions can only occur when the Gaussian curvature on surface is negative or zero along the asymptotic curve [1, 4, 5].
From the definitions of the conjugate vectors, the asymptotic directions and the equation II*(X; Y) = II(X; Y), we say the conjugate vectors and asymptotic directions in M are also the conjugate vectors and asymptotic directions in M*.
(iii) Asymptotic directions in M are also the asymptotic directions in M*.

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