Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far. This work introduces the unified theoretic framework for discrete surface Ricci flow, including all the common schemes: tangential circle packing, Thurston’s circle packing, inversive distance circle packing and discrete Yamabe flow. Furthermore, this work also introduces a novel schemes, virtual radius circle packing and the mixed type schemes, under the unified framework. This work gives explicit geometric interpretation to the discrete Ricci energies for all the schemes with all back ground geometries, and the corresponding Hessian matrices. The unified frame work deepens our understanding to the discrete surface Ricci flow theory, and has inspired us to discover the new schemes, improved the flexibility and robustness of the algorithms, greatly simplified the implementation and improved the efficiency. Experimental results show the unified surface Ricci flow algorithms can handle general surfaces with different topologies, and is robust to meshes with different qualities, and is effective for solving real problems. 
The unified discrete surface Ricci flow