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.
 |    |
