Tutorial on Discrete Ricci Flow for Global Parameterizations

Xianfeng Gu, Feng Luo, Shing-Tung Yau
This work introduces the concepts and methods for Ricci flow for computer scientists and engineers. Readers can understand the background theories as well as the implementation details, such that they can make Ricci flow software easily and find potential applications in graphics field. First, the basic concepts from local differential geometry are briefly introduced, the concepts of metric, curvature are explained in details. Then different energies are defined to quantitative measure the distorison of parameterizations. The conformal parameterizations are emphasized. Second, the theories from global differential geometry are thoroughly explained, such as manifolds, affine atlas, Riemann surfaces, Riemann uniformization theorem. Then Ricci flow is introduced to conformally deform surfaces, such that the solution surfaces have constant Gaussian curvatures. Third, the concepts and methods from continuous geometry are systematically translated to the discrete setting via circle packing metric. The discrete Ricci flow is thoroughly explained, the existence of the solution, the exponential convergence, the variational energy, the Newton¡¯s method are explained. Finally, discrete Ricci flow is implemented based a common mesh library. The details of the algorithms are illuminated. Experimental results are illustrated and discussed. Readers who are only interested in the implementation of Ricci flow can skip the first two chapters.