Loading AI tools
From Wikipedia, the free encyclopedia
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.
There are several equivalent definitions for asymptotic directions, or equivalently, asymptotic curves.
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 directions. There is one or infinitely many asymptotic directions through every point with zero Gaussian curvature.
If the surface is 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 generatrices, and them only.
If a straight line is included in a surface, then it is an asymptotic curve of the surface.
A related notion is a curvature line, which is a curve always tangent to a principal direction.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.