In the
differential geometry of surfaces, 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 (geometry), line, but that does not have to be Linearity, straight.
Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
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. More ...
to an asymptotic direction of the surface (where they exist). It is sometimes called an asymptotic line, although it need not be a
line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ...
.
Definitions
An asymptotic direction is one in which the normal
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
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 (gen ...
which bears both the curve's tangent and the surface's
normal 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
In differential geometry, the Gaussian curvature or Gauss curvature of a surface at a point is the product of the principal curvatures, and , at the given point:
K = \kappa_1 \kappa_2.
The Gaussian radius of curvature is the reciprocal of .
F ...
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
asymptotes of the hyperbola of the
Dupin indicatrix.
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
French (french: français(e), link=no) may refer to:
* Something of, from, or related to France
** French language, which originated in France, and its various dialects and accents
** French people, a nation and ethnic group identified with Franc ...
)
Curves
Differential geometry of surfaces
Surfaces
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