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)
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where , , and are coefficients of the second fundamental form. The differential equation for asymptotic curves on a Monge patch is
(2)
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and on a polar patch is
(3)
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The images below show asymptotic curves for the elliptic helicoid, funnel, hyperbolic paraboloid, and monkey saddle.
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