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.