Classification of surface points

The Gauss map at a point can be used to classify the local geometry by archetypal geometries. A point is called

• elliptic if $det(dN_p) > 0$ (e.g. a sphere),
• hyperbolic if $det(dN_p) < 0$ (e.g. a saddle),
• parabolic if $det(dN_p) = 0$ and $dN_p \neq 0$ (e.g. a cylinder), and
• planar if $det(dN_p) = 0$ .