Asymptotic curve
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
.
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
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
is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are bisected by the principal directions
.
The direction of the asymptotic direction are the same as the asymptote
s of the hyperbola of the Dupin indicatrix
.
A related notion is a curvature line
, which is a curve always tangent to a principal direction.
Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with smooth surfaces with various additional structures, most often, a Riemannian metric....
, an asymptotic curve is a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
always tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...
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 (mathematics)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
.
An asymptotic direction is one in which the normal curvature
Curvature
In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
is zero. Which is to say: for a point on an asymptotic curve, take the plane
Plane (mathematics)
In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
which bears both the curve's tangent and the surface's normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
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
Gaussian curvature
In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...
is negative (or zero). There will be two asymptotic directions through every point with negative Gaussian curvature, these directions are bisected by the principal directions
Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point. They measure how the surface bends by different amounts in different directions at that point.-Discussion:...
.
The direction of the asymptotic direction are the same as the asymptote
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 they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
s of the hyperbola of the Dupin indicatrix
Dupin indicatrix
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. The shape of the intersection is related to the Gaussian curvature. The Dupin...
.
A related notion is a curvature line
MonStAR
"MonStAR" is a CD single by Japanese singer and voice actress Aya Hirano. It was released on December 5, 2007 and was produced by Lantis. This is Aya Hirano's sixth maxi single and her third and last release from her three-month consecutive single release campaign.-Track listing:# "MonStAR"#*...
, which is a curve always tangent to a principal direction.