  Quadratic points of a surface in the projective 3-space are the points which
can be exceptionally well approximated by a quadric. They are also
singularities of a 3-web in the elliptic part and of a line field in the
hyperbolic part of the surface. We show that generically the index of the 3-web
at a quadratic point is 1/3 or -1/3, while the index of the line field is 1 or
-1. Moreover, for an elliptic quadratic point whose cubic form is
semi-homogeneous, we can use Loewner's conjecture to show that the index is at
most 1.
  From the above local results we can conclude some global results: A generic
compact elliptic surface has at least 6 quadratic points, a compact elliptic
surfaces with semi-homogeneous cubic forms has at least 2 quadratic points and
the number of quadratic points in a hyperbolic disc is odd. By studying the
behavior of the cubic form in a neighborhood of the parabolic curve, we also
obtain a relation between the indices of the quadratic points of a generic
surface with non-empty elliptic and hyperbolic regions.
