A Geometric Representation of Improper Indefinite Affine Spheres with Singularities

TL;DR

This paper establishes a geometric link between improper indefinite affine spheres with singularities and the generalized area distance of planar curves, providing a new perspective on their structure and properties.

Contribution

It demonstrates that improper indefinite affine spheres with singularities can be characterized as generalized area distances of planar curve pairs, revealing a reciprocal relationship.

Findings

Improper indefinite affine spheres correspond to generalized area distances of planar curves.

Singularities of the affine spheres relate to the area evolute of the curve pairs.

Geometric properties of the spheres can be described using symmetry sets of the curves.

Abstract

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.