# MathSciDoc: An Archive for Mathematician ∫

#### Computational GeometryTBDmathscidoc:1911.43001

Computer Methods in Applied Mechanics and Engineering, 356, 652-668, 2019
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.
Quadrilateral mesh; Flat Riemannian metric; Geodesic; Discrete Ricci flow; Conformal structure deformation