Mathematics – Differential Geometry
Scientific paper
2010-12-09
Mathematics
Differential Geometry
14 pages 7 figures
Scientific paper
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.
Craizer Marcos
da Silva Moacyr A. H. B.
Teixeira Ralph C.
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