Asymptotic curve

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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.

Contents

Definitions

There are several equivalent definitions for asymptotic directions, or equivalently, asymptotic curves.

Properties

Asymptotic directions can only occur when the Gaussian curvature is negative (or zero).

There are two asymptotic directions through every point with negative Gaussian curvature, bisected by the principal directions. There is one or infinitely many asymptotic directions through every point with zero Gaussian curvature.

If the surface is minimal and not flat, then the asymptotic directions are orthogonal to one another (and 45 degrees with the two principal directions).

For a developable surface, the asymptotic lines are the generatrices, and them only.

If a straight line is included in a surface, then it is an asymptotic curve of the surface.

A related notion is a curvature line, which is a curve always tangent to a principal direction.

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References

  1. David Hilbert; Cohn-Vossen, S. (1999). Geometry and Imagination. American Mathematical Society. ISBN   0-8218-1998-4.