In the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
, 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 the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
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
An asymptotic direction is one in which the normal
curvature is zero. Which is to say: for a point on an asymptotic curve, take the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
which bears both the curve's tangent and the surface's
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
at that point. The curve of intersection of the plane and the surface will have zero curvature at that point. Asymptotic directions can only occur when the
Gaussian curvature is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, bisected by the
principal directions. If the surface is
minimal, the asymptotic directions are orthogonal to one another.
Related notions
The direction of the asymptotic direction 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 context ...
s of the hyperbola of the
Dupin indicatrix
In differential geometry, the Dupin indicatrix is a method for characterising the local shape of a surface. Draw a plane parallel to the tangent plane and a small distance away from it. Consider the intersection of the surface with this plane. Th ...
.
A related notion is a
curvature line
In differential geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues of the shape operator at that point. They measure how the surface bends by d ...
, 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
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