Asymptotic Curve

Given a regular surface M, an asymptotic curve is formally defined as a curve x(t) on M such that the normal curvature is 0 in the direction x^'(t) for all t in the domain of x. The differential equation for the parametric representation of an asymptotic curve is


where e, f, and g are coefficients of the second fundamental form. The differential equation for asymptotic curves on a Monge patch (u,v,h(u,v)) is


and on a polar patch (rcostheta,rsintheta,h(r)) is


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


See also

Ruled Surface

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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.

Referenced on Wolfram|Alpha

Asymptotic Curve

Cite this as:

Weisstein, Eric W. "Asymptotic Curve." From MathWorld--A Wolfram Web Resource.

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