# asymptotic curve

(redirected from Asymptotic direction)

## asymptotic curve

[ā‚sim′täd·ik ′kərv]
(mathematics)
A curve on a surface whose osculating plane at each point is the same as the tangent plane to the surface.
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Further articulated in part two, it proposes a systematic argument for the contention that "truth is intention, telos, asymptotic direction" (p.
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].
If II ([X.sub.P], [X.sub.P]) = 0, then XP is called the asymptotic direction. The fundamental form [I.sup.q], 1 [less than or equal to] q [less than or equal to] 3, on M such that
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*.
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 ).

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