In the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth manifold, smooth Surface (topology), surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensiv ...
, an asymptotic curve is a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
always
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a
line.
Definitions
There are several equivalent definitions for asymptotic directions, or equivalently, asymptotic curves.
* The asymptotic directions are the same as the
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
s of the hyperbola of the
Dupin indicatrix through a
hyperbolic point, or the unique asymptote through a
parabolic point.
* An asymptotic direction is a direction along which the normal
curvature
In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
is zero: take the
plane spanned by the direction and the surface's
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
In differential geometry, the Gaussian curvature or Gauss curvature of a smooth Surface (topology), surface in three-dimensional space at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
For ...
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
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...
, 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.
Related notions
A related notion is a
curvature line, which is a curve always tangent to a principal direction.
References
* {{MathWorld , urlname=AsymptoticCurve , title=Asymptotic Curve
Lines of Curvature, Geodesic Torsion, Asymptotic Lines"Asymptotic line of a surface" at Encyclopédie des Formes Mathématiques Remarquables(in
French)
Curves
Differential geometry of surfaces
Surfaces