In the differential geometry of surfaces, an asymptotic curve is a curve always tangent to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line although it needn't be a line.

An asymptotic direction is one in which the normal curvature is zero. Which is to say: for a point on an asymptotic curve, take the plane which bears both the curve's tangent and the surface's normal at that point. The curve of intersection of the plane and the surface will have zero curvature at that point.

This is quite distinct from an asymptote; asymptotic curves are embedded in their surface whereas asymptotes may be completely external to their figures.

This is also not the same as a geodesic.

References

• Mathworld
• [1]
• Mathcurve (in French)

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