caa a22 4500 445813288 CHVBK 20180317145218.0 cr unu---uuuuu 170323e20110401xx s 000 0 eng 10.1007/s00022-011-0078-y doi (NATIONALLICENCE)springer-10.1007/s00022-011-0078-y A geometric representation of improper indefinite affine spheres with singularities [Elektronische Daten] [Marcos Craizer, Ralph Teixeira, Moacyr da Silva] 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 $${\mathbb {R}^3}$$ 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. Springer Basel AG, 2011 Improper affine spheres nationallicence area distances nationallicence affine symmetry sets nationallicence Craizer Marcos Departamento de Matemática-PUC-Rio, Rio de Janeiro, Brazil aut Teixeira Ralph Departamento de Matemática Aplicada-UFF, Niterói, Brazil aut da Silva Moacyr Centro de Matemática Aplicada-FGV, Rio de Janeiro, Brazil aut Journal of Geometry SP Birkhäuser Verlag Basel 100/1-2(2011-04-01), 65-78 0047-2468 100:1-2<65 2011 100 22 https://doi.org/10.1007/s00022-011-0078-y text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00022-011-0078-y text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Craizer Marcos Departamento de Matemática-PUC-Rio, Rio de Janeiro, Brazil aut NATIONALLICENCE 700 1- Teixeira Ralph Departamento de Matemática Aplicada-UFF, Niterói, Brazil aut NATIONALLICENCE 700 1- da Silva Moacyr Centro de Matemática Aplicada-FGV, Rio de Janeiro, Brazil aut NATIONALLICENCE 773 0- Journal of Geometry SP Birkhäuser Verlag Basel 100/1-2(2011-04-01), 65-78 0047-2468 100:1-2<65 2011 100 22 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer