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A curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137101.png" /> on a regular surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137102.png" /> such that the normal curvature along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137103.png" /> is zero. An asymptotic line is given by the differential equation:
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137104.png" /></td> </tr></table>
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A curve  $  \Gamma $
 +
on a regular surface  $  F $
 +
such that the normal curvature along  $  \Gamma $
 +
is zero. An asymptotic line is given by the differential equation:
 +
 
 +
$$
 +
\textrm{ II }  = L  du  ^ {2} +2M  du  dv + N  dv  ^ {2}  = 0 ,
 +
$$
  
 
where II is the second fundamental form of the surface.
 
where II is the second fundamental form of the surface.
  
The osculating plane of an asymptotic line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137105.png" />, if it exists, coincides with the tangent plane to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137106.png" /> (at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137107.png" />), and the square of the torsion of an asymptotic line is equal to the modulus of the Gaussian curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137108.png" /> of the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a0137109.png" /> (the Beltrami–Enneper theorem). A straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371010.png" /> (e.g. a generating line of a ruled surface) is always an asymptotic line. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371011.png" /> is a parabolic curve (e.g. a circle on a standard torus separating the domains of Gaussian curvatures of different signs), it is an asymptotic line.
+
The osculating plane of an asymptotic line $  \Gamma $,  
 +
if it exists, coincides with the tangent plane to $  F $(
 +
at the points of $  \Gamma $),  
 +
and the square of the torsion of an asymptotic line is equal to the modulus of the Gaussian curvature $  K $
 +
of the surface $  F $(
 +
the Beltrami–Enneper theorem). A straight line $  l \in F $(
 +
e.g. a generating line of a ruled surface) is always an asymptotic line. If $  \Gamma $
 +
is a parabolic curve (e.g. a circle on a standard torus separating the domains of Gaussian curvatures of different signs), it is an asymptotic line.
  
A unique asymptotic line, which coincides with the rectilinear generator, passes through each point of a parabolic domain (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371012.png" />, but II<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371013.png" />). Through each point of a hyperbolic domain (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371014.png" />) there pass exactly two asymptotic lines, forming the so-called [[Asymptotic net|asymptotic net]], which plays an important role in the study of the spatial form of a surface of negative curvature (cf. [[Negative curvature, surface of|Negative curvature, surface of]]). For instance, on a complete surface this net is homeomorphic to the Cartesian net on the plane if
+
A unique asymptotic line, which coincides with the rectilinear generator, passes through each point of a parabolic domain (where $  K = 0 $,  
 +
but II $  \neq 0 $).  
 +
Through each point of a hyperbolic domain (where $  K<0 $)  
 +
there pass exactly two asymptotic lines, forming the so-called [[Asymptotic net|asymptotic net]], which plays an important role in the study of the spatial form of a surface of negative curvature (cf. [[Negative curvature, surface of|Negative curvature, surface of]]). For instance, on a complete surface this net is homeomorphic to the Cartesian net on the plane if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371015.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm grad} \left |
 +
\frac{1} \sqrt
 +
-K \right |  \leq  q,\ \
 +
q = \textrm{ const } .
 +
$$
  
Asymptotic nets on surfaces of constant negative curvature are Chebyshev nets (cf. [[Chebyshev net|Chebyshev net]]), and the surface area of a quadrangle formed by asymptotic lines is proportional to the excess of the sum of its interior angles <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371017.png" />:
+
Asymptotic nets on surfaces of constant negative curvature are Chebyshev nets (cf. [[Chebyshev net|Chebyshev net]]), and the surface area of a quadrangle formed by asymptotic lines is proportional to the excess of the sum of its interior angles $  \alpha _ {i} $
 +
over $  2 \pi $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371018.png" /></td> </tr></table>
+
$$
 +
| K | S  = 2 \pi - \alpha _ {1} - \alpha _ {2} - \alpha _ {3} -
 +
\alpha _ {4}  $$
  
 
(Hazzidakis' formula).
 
(Hazzidakis' formula).
  
Under a projective transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371019.png" /> of the space, the asymptotic lines of a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371020.png" /> become the asymptotic lines of the transformed surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013710/a01371021.png" />.
+
Under a projective transformation $  \pi $
 +
of the space, the asymptotic lines of a surface $  F $
 +
become the asymptotic lines of the transformed surface $  \pi (F) $.
  
 
Asymptotic lines on surfaces in a three-dimensional Riemannian space are defined in a similar manner. Various generalizations of the concept of an asymptotic line on manifolds imbedded in a multi-dimensional space are known; the most frequently-used one involves the concept of the second fundamental form, which is associated with a given normal vector.
 
Asymptotic lines on surfaces in a three-dimensional Riemannian space are defined in a similar manner. Various generalizations of the concept of an asymptotic line on manifolds imbedded in a multi-dimensional space are known; the most frequently-used one involves the concept of the second fundamental form, which is associated with a given normal vector.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Rashevskii,  "A course of differential geometry" , Moscow  (1956)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.V. Pogorelov,  "Differential geometry" , Noordhoff  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  P.K. Rashevskii,  "A course of differential geometry" , Moscow  (1956)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 18:48, 5 April 2020


A curve $ \Gamma $ on a regular surface $ F $ such that the normal curvature along $ \Gamma $ is zero. An asymptotic line is given by the differential equation:

$$ \textrm{ II } = L du ^ {2} +2M du dv + N dv ^ {2} = 0 , $$

where II is the second fundamental form of the surface.

The osculating plane of an asymptotic line $ \Gamma $, if it exists, coincides with the tangent plane to $ F $( at the points of $ \Gamma $), and the square of the torsion of an asymptotic line is equal to the modulus of the Gaussian curvature $ K $ of the surface $ F $( the Beltrami–Enneper theorem). A straight line $ l \in F $( e.g. a generating line of a ruled surface) is always an asymptotic line. If $ \Gamma $ is a parabolic curve (e.g. a circle on a standard torus separating the domains of Gaussian curvatures of different signs), it is an asymptotic line.

A unique asymptotic line, which coincides with the rectilinear generator, passes through each point of a parabolic domain (where $ K = 0 $, but II $ \neq 0 $). Through each point of a hyperbolic domain (where $ K<0 $) there pass exactly two asymptotic lines, forming the so-called asymptotic net, which plays an important role in the study of the spatial form of a surface of negative curvature (cf. Negative curvature, surface of). For instance, on a complete surface this net is homeomorphic to the Cartesian net on the plane if

$$ \mathop{\rm grad} \left | \frac{1} \sqrt -K \right | \leq q,\ \ q = \textrm{ const } . $$

Asymptotic nets on surfaces of constant negative curvature are Chebyshev nets (cf. Chebyshev net), and the surface area of a quadrangle formed by asymptotic lines is proportional to the excess of the sum of its interior angles $ \alpha _ {i} $ over $ 2 \pi $:

$$ | K | S = 2 \pi - \alpha _ {1} - \alpha _ {2} - \alpha _ {3} - \alpha _ {4} $$

(Hazzidakis' formula).

Under a projective transformation $ \pi $ of the space, the asymptotic lines of a surface $ F $ become the asymptotic lines of the transformed surface $ \pi (F) $.

Asymptotic lines on surfaces in a three-dimensional Riemannian space are defined in a similar manner. Various generalizations of the concept of an asymptotic line on manifolds imbedded in a multi-dimensional space are known; the most frequently-used one involves the concept of the second fundamental form, which is associated with a given normal vector.

References

[1] A.V. Pogorelov, "Differential geometry" , Noordhoff (1959) (Translated from Russian)
[2] P.K. Rashevskii, "A course of differential geometry" , Moscow (1956) (In Russian)

Comments

Hazzidakis' formula can be found in [a1] and [a2], p. 204.

References

[a1] J.N. Hazzidakis, "Uber einige Eigenschaften der Flächen mit konstanten Krümmungsmasz" Crelle's J. Math. , 88 (1880) pp. 68–73
[a2] D.J. Struik, "Lectures on classical differential geometry" , Addison-Wesley (1950)
[a3] C.C. Hsiung, "A first course in differential geometry" , Wiley (1981) pp. Chapt. 3, Sect. 4
[a4] M. Spivak, "A comprehensive introduction to differential geometry" , 3 , Publish or Perish (1975) pp. 1–5
[a5] N.J. Hicks, "Notes on differential geometry" , v. Nostrand (1965)
[a6] W. Blaschke, K. Leichtweiss, "Elementare Differentialgeometrie" , Springer (1973)
How to Cite This Entry:
Asymptotic line. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Asymptotic_line&oldid=12030
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article