Asymptotic Direction

An asymptotic direction at a point p of a Regular Surface $M\in\Bbb{R}^3$ is a direction in which the Normal Curvature of $M$ vanishes.

1. There are no asymptotic directions at an Elliptic Point.

2. There are exactly two asymptotic directions at a Hyperbolic Point.

3. There is exactly one asymptotic direction at a Parabolic Point.

4. Every direction is asymptotic at a Planar Point.

See also Asymptotic Curve

References

Gray, A. Modern Differential Geometry of Curves and Surfaces.Boca Raton, FL: CRC Press, pp. 270 and 320, 1993.