Braid Monodromy Type and Rational Transformations of Plane Algebraic   Curves
  We combine the newly discovered technique, which computes explicit formulas for the image of an algebraic curve under rational transformation, with techniques that enable to compute braid monodromies of such curves. We use this combination in order to study properties of the braid monodromy of the image of curves under a given rational transformation. A description of the general method is given along with full classification of the images of two intersecting lines under degree 2 rational transformation. We also establish a connection between degree 2 rational transformations and the local braid monodromy of the image at the intersecting point of two lines. Moreover, we present an example of two birationally isomorphic curves with the same braid monodromy type and non diffeomorphic real parts. 
S. Kaplan, A. Shapiro and M. Teicher