# Asymptotic line

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)