Asymptotic Curve

Given a Regular Surface $M$, an asymptotic curve is formally defined as a curve ${\bf x}(t)$ on $M$ such that the Normal Curvature is 0 in the direction ${\bf x}'(t)$ for all $t$ in the domain of ${\bf x}$. The differential equation for the parametric representation of an asymptotic curve is

$$eu'^2+2fu'v'+gv'^2=0,$$ (1)

where $e$, $f$, and $g$ are second Fundamental Forms. The differential equation for asymptotic curves on a Monge Patch $(u, v, h(u,v))$ is

$$h_{uu}u'^2+2h_{uu}u'v'+h_{vv}v'^2=0,$$ (2)

and on a polar patch $(r\cos\theta, r\sin\theta, h(r))$ is

$$h''(r)r'^2+h'(r)r\theta'^2=0.$$ (3)

The images below show asymptotic curves for the Elliptic Helicoid, Funnel, Hyperbolic Paraboloid, and Monkey Saddle.

See also Ruled Surface

References

Gray, A. ``Asymptotic Curves,'' ``Examples of Asymptotic Curves,'' ``Using Mathematica to Find Asymptotic Curves.'' §16.1, 16.2, and 16.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 320-331, 1993.