The area distance to a convex plane curve is an important concept in computer
vision. In this paper we describe a strong link between area distances and
improper affine spheres. This link makes possible a better understanding of
both theories. The concepts of the theory of affine spheres lead to a new
definition of an area distance on the outer part of a convex plane arc. Also,
based on the theory of discrete affine spheres, we propose fast algorithms to
compute the area distances. On the other hand, area distances provide a good
geometrical understanding of improper affine spheres.Comment: 14 pages 18 figure
