info prev up next book cdrom email home

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

\end{displaymath} (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
\end{displaymath} (2)

and on a polar patch $(r\cos\theta, r\sin\theta, h(r))$ is
\end{displaymath} (3)

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

\begin{figure}\begin{center}\BoxedEPSF{EllipticalHelicoidAsymp.epsf scaled 490}\...
...d 490}\quad\BoxedEPSF{MonkeySaddleAsymp.epsf scaled 490}\end{center}\end{figure}

See also Ruled Surface


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.

© 1996-9 Eric W. Weisstein