In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist).
Asymptotic directions can only occur when the Gaussian curvature is negative (or zero).
There is one or infinitely many asymptotic directions through every point with zero Gaussian curvature.
For a developable surface, the asymptotic lines are the generatrices, and them only.
A related notion is a curvature line, which is a curve always tangent to a principal direction.