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# Asymptotic Curve

Given a regular surface , an asymptotic curve is formally defined as a curve on such that the normal curvature is 0 in the direction for all in the domain of . The differential equation for the parametric representation of an asymptotic curve is

 (1)

where , , and are coefficients of the second fundamental form. The differential equation for asymptotic curves on a Monge patch is

 (2)

and on a polar patch is

 (3)

The images below show asymptotic curves for the elliptic helicoid, funnel, hyperbolic paraboloid, and monkey saddle.

Ruled Surface

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## References

Gray, A. "Asymptotic Curves," "Examples of Asymptotic Curves," and "Using Mathematica to Find Asymptotic Curves." §18.1, 18.2, and 18.3 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 417-429, 1997.

Asymptotic Curve

## Cite this as:

Weisstein, Eric W. "Asymptotic Curve." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AsymptoticCurve.html