  We introduce a new family of affine metrics on a locally strictly convex
surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric
equiaffine planes associated with each metric. We show that if $M$ is immersed
in a locally strictly convex hyperquadric, then the symmetric and the
antisymmetric planes coincide and contain the affine normal to the
hyperquadric. In particular, any surface immersed in a locally strictly convex
hyperquadric is affine semiumbilical with respect to the symmetric or
antisymmetric equiaffine planes.
