Since the normal curvature of a surface at a regular point is a continuous function, it has both a maximum and a minimum on (according to the extreme value theorem). These extrema are the principal curvatures and of at and the Euler curvature formula is valid. If , defines the asymptotic direction at .
The Gaussian curvature of at is defined as .
A point on the surface is called:
- An elliptic point if (or the principal curvatures and have the same sign). There are no asymptotic directions.
- A hyperbolic point if (or the principal curvatures and have opposite signs). There are exactly two asymptotic directions, and the principal directions bisect the angle between them.
- A parabolic point if and exactly one of the principal curvatures vanishes.
- A planar point if and both and vanish—that is, every direction is asymptotic.