Algorithm 3.

Embed the surface onto the Poincaré disk.

	Input: A multiply connected surface Σ with its hyperbolic uniformization metric and   its simply connected domain Σ̃.
	Output: The embedding of Σ̃ onto the Poincaré disk.
1	Copy the hyperbolic uniformization metric from Σ to Σ̃.
2	Select a seed face f012 = [υ0, υ1, υ2] ∈ Σ̃, map each vertex to the Poincaré disk as: p(υ0) = (0, 0), $p({\upsilon }_1)=\frac{e^{l_{01}}-1}{e^{l_{01}}+1}(1,0)$, and $p({\upsilon }_2)=\frac{e^{l_{02}}-1}{e^{l_{02}}+1}(\text{cos}{\theta }_0^{12},\text{sin}{\theta }_0^{12})$.
3	Put all the neighboring faces of f012 to a queue.
4	while the queue is not empty do
5		Pop the first face fijk from the queue;
6		if fijk has been embedded then
7			Continue;
8		else
9			Suppose υi and υj have been embedded, compute the intersections of two circles (center, radius): (p(υi), lik) and (p(υj), ljk).
10			Then p(υk) is chosen as the intersection point that keeps the face upward.
11			Put the neighboring faces of fijk in the queue.
12		end
13	end