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.
 |    |
