Algorithm 2.

Hyperbolic Ricci Flow with Newton’s Optimization

	Input: A triangular mesh Σ(V, E, F).
	Output: Hyperbolic metric of Σ.
1	Compute the initial ri for each vertex υi and the weight ϕij for each edge eij.
2	Set the target Gaussian curvature as zero.
3	repeat Compute the edge lengths with Eq. 5, face corner angles with Eq. 3, and the Gaussian curvature with Eq. 4. For each vertex υi, compute $\frac{\partial K_i}{\partial u_i}$ and $\frac{\partial K_i}{\partial u_j}$ with Eq. 9 and construct the Hessian matrix H. Solve the linear system HΔu = −2K. Update ui for vertex υi with ui ← ui + Δui. Update ri for vertex υi with Eq. 6.
	until the resulting Gaussian curvature of all vertices is less than a user-defined threshold.