Quad meshing is a fundamental preprocessing task for many applications (subdivision surfaces, boundary layer simulation). State-of-the-art quad mesh generators proceed in three steps: first a guiding cross field is computed, then a parametrization representing the quads is generated, and finally a mesh is extracted from the parameterization. In this paper we show that in the case of a periodic global parameterization two first steps answer to the same equation and inherently face the same challenges. This new insight allows us to use recent cross field generation algorithms based on Ginzburg-Landau equations to accurately solve the parametrization step. We provide practical evidence that this formulation enables us to overcome common shortcomings in parametrization computation (inaccuracy away from the boundary, singular dipole placement).