View source for Asymptotic line

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:

<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>

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.

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

<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>

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" />:

<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>

(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" />.

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====
<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====
Hazzidakis' formula can be found in [[#References|[a1]]] and [[#References|[a2]]], p. 204.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.N. Hazzidakis,   "Uber einige Eigenschaften der Flächen mit konstanten Krümmungsmasz"  ''Crelle's J. Math.'' , '''88'''  (1880)  pp. 68–73</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.J. Struik,   "Lectures on classical differential geometry" , Addison-Wesley  (1950)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.C. Hsiung,   "A first course in differential geometry" , Wiley  (1981)  pp. Chapt. 3, Sect. 4</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Spivak,   "A comprehensive introduction to differential geometry" , '''3''' , Publish or Perish  (1975)  pp. 1–5</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N.J. Hicks,   "Notes on differential geometry" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  W. Blaschke,   K. Leichtweiss,   "Elementare Differentialgeometrie" , Springer  (1973)</TD></TR></table>