In a recent work, a new indirect method to generate all-quad meshes has been developed. It takes advantage of a well-known algorithm of the graph theory, namely the Blossom algorithm, which computes in polynomial time the minimum cost perfect matching in a graph. In this paper, we describe a method that allows to build triangular meshes that are better suited for recombination into quadrangles. This is performed by using the infinity norm to compute distances in the meshing process. The alignment of the elements in the frontal Delaunay procedure is controlled by a cross field defined on the domain. Meshes constructed this way have their points aligned with the cross-field directions, and their triangles are almost right everywhere. Then, recombination with the Blossom-based approach yields quadrilateral meshes of excellent quality. Copyright © 2013 John Wiley & Sons, Ltd.