This work proposes a novel metric based algorithm for quadrilateral mesh generating. Each quad-mesh induces a Riemannian
metric satisfying special conditions: the metric is a flat metric with cone singularities conformal to the original metric, the
total curvature satisfies the Gauss–Bonnet condition, the holonomy group is a subgroup of the rotation group $\{e^{ik\pi/2}\}$, there
is cross field obtained by parallel translation which is aligned with the boundaries, and its streamlines are finite geodesics.
Inversely, such kind of metric induces a quad-mesh. Based on discrete Ricci flow and conformal structure deformation, one
can obtain a metric satisfying all the conditions and obtain the desired quad-mesh.
This method is rigorous, simple and automatic. Our experimental results demonstrate the efficiency and efficacy of the
algorithm.