Gaussian image of a surface

We consider a surface curve containing a given point on a surface, and the associated Gaussian image, in the neighbourhood of a

• elliptic point: The curve and the Gaussian image have the same orientation.

• hyperbolic point: The curve and the Gaussian image have opposite orientations.

• parabolic point: The Gaussian image of the curve has a double point.

• "monkey saddle:" The Gaussian image has a different winding number.

Reference: Hilbert/Cohn-Vossen: "Geometry and the Imagination" (translation from the German "Anschauliche Geometrie").