Editing Asymptotic curve

In the [[differential geometry of surfaces]], an '''asymptotic curve''' is a [[curve]] always [[tangent]] to an '''asymptotic direction''' of the surface (where they exist). It is sometimes called an '''asymptotic line''', although it need not be a [[line (mathematics)|line]]. 

==Definitions==
There are several equivalent definitions for asymptotic directions, or equivalently, asymptotic curves.

* The asymptotic directions are the same as the [[asymptote]]s of the hyperbola of the [[Dupin indicatrix]] through a [[Principal_curvature#Classification of points on a surface| hyperbolic point]], or the unique asymptote through a [[Principal_curvature#Classification of points on a surface| parabolic point]].<ref>{{cite book |author=David Hilbert |title=Geometry and Imagination |author2=Cohn-Vossen, S. |publisher=American Mathematical Society |year=1999 |isbn=0-8218-1998-4 |authorlink=David Hilbert |authorlink2=Stephan Cohn-Vossen}}</ref>

* An asymptotic direction is a direction along which the normal [[curvature]] is zero: take the [[plane (mathematics)|plane]] spanned by the direction and the surface's [[surface normal|normal]] at that point. The curve of intersection of the plane and the surface has zero curvature at that point.
* An asymptotic curve is a curve such that, at each point, the plane tangent to the surface is an osculating plane of the curve.

== Properties ==
Asymptotic directions can only occur when the [[Gaussian curvature]] is negative (or zero).  

There are two asymptotic directions through every point with negative Gaussian curvature, bisected by the [[principal curvature|principal directions]]. There is one or infinitely many asymptotic directions through every point with zero Gaussian curvature. 

If the surface is [[Minimal surface|minimal]] and not flat, then the asymptotic directions are orthogonal to one another (and 45 degrees with the two principal directions). 

For a [[developable surface]], the asymptotic lines are the [[Generatrix|generatrices]], and them only. 

If a straight line is included in a surface, then it is an asymptotic curve of the surface.

==Related notions==
A related notion is a [[Principal_curvature#Line_of_curvature|curvature line]], which is a curve always tangent to a principal direction.

==References==
{{Reflist}}

* {{MathWorld |urlname=AsymptoticCurve |title=Asymptotic Curve}}
* [https://web.archive.org/web/20040919132201/http://www.seas.upenn.edu/~cis70005/cis700sl10pdf.pdf Lines of Curvature, Geodesic Torsion, Asymptotic Lines]
* [http://www.mathcurve.com/surfaces/asymptotic/asymptotic.shtml "Asymptotic line of a surface" at Encyclopédie des Formes Mathématiques Remarquables] (in [[French language|French]])

[[Category:Curves]]
[[Category:Differential geometry of surfaces]]
[[Category:Surfaces]]