Abstract
Gu, Luo, and Yau etc. provided a process to discretize the Ricci flow and find the conformal mapping in 2007. In this thesis, we introduce the method and implement its algorithms to embed a closed genus-1 surface (in 3-D) into the Euclidean plane conformally. The algorithm is composed of two parts. First, we find the solution of the Ricci flow with Newton's method. By the convergence theory, it is known that there is a solution of the Ricci equation. We can write the equation as a time-independent equation system which solved by Newton's method. Newton's method is excellent on the rate of convergence. Second, we implement an efficiency way by Yau and Gu to find the position of vertices on $\mathbb{R}^2$ since the iterative method is intuitive but inefficient. Hence, the efficient method brings more benefits. Next, we use two methods to estimate the conformal errors. Finally, we provide some advices about the whole procedure to find the conformal mapping.