In 1984, I was able to obtain a parametrization of the Boy surface by eliminating the Whitney umbrellas of the Steiner surface using the so-called hyperbolic confluence of pairs of singularities. As a result, the Boy surface appears to be a real algebraic surface of degree six. However, the construction was a mix of geometry, differential topology and singularity theory. In this talk I want to investigate the question in the complex algebraic geometry field. We intend to characterize the Boy surface as an complex algebraic surface subjected to natural conditions.