Editing Asymptotic curve (section)

==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.