  Given a pair of planar curves, one can define its generalized area distance,
a concept that generalizes the area distance of a single curve. In this paper,
we show that the generalized area distance of a pair of planar curves is an
improper indefinite affine spheres with singularities, and, reciprocally, every
indefinite improper affine sphere in $\R^3$ is the generalized distance of a
pair of planar curves. Considering this representation, the singularity set of
the improper affine sphere corresponds to the area evolute of the pair of
curves, and this fact allows us to describe a clear geometric picture of the
former. Other symmetry sets of the pair of curves, like the affine area
symmetry set and the affine envelope symmetry set can be also used to describe
geometric properties of the improper affine sphere.
