Gruber P. Convex and Discrete Geometry

510 Geometry of Numbers
Thurston’s Algorithm
Fig. 34.6. Circle packing corresponding to a graph
We need the following special case of Theorem 34.1:
(1) Let G be a planar, 3-connected graph all countries of which are triangular,
including the exterior country. Then to G corresponds a packing of (circular)
discs in C. The discs are in one-to-one correspondence with the vertices
of G and two discs touch precisely when the corresponding vertices are
connected by an edge (Fig. 34.6).
Let v1,...,vn be the vertices of G where v1,v2,v3 are the vertices of the exterior
country. For the construction of the corresponding packing of discs, it is sufficient to
specify the radii ϱ1,...,ϱn of the discs corresponding to v1,...,vn (up to Möbius
transformations). The algorithm can be described as follows:
For k = 1, 2,...,assign to v1,...,vn labels ϱk1,...,ϱkn in the following way:
Let ϱ11 = ϱ12 = ϱ13 = 1 and ϱ14,...,ϱ1n > 0 arbitrary. Now cycle through
v4,...,vn,vn+1 = v4. Assume that at step k we have arrived at vertex vi−1. If
viv jvk is a (triangular) country of G, its angle at the vertex vi is the angle of the
Euclidean triangle with edges of lengths ϱki + ϱkj,ϱki + ϱkl,ϱkj + ϱkl opposite to
the edge of length ϱkj + ϱkl. Thecurvature κ(vi,ϱk1,...,ϱkn) at vi is 2π minus
the sum of the angles at vi of the triangular countries with vertex vi. Considered as a
function of ϱ = ϱki, the curvature is strictly increasing. If ϱ →+∞, the curvature at
vi tends to 2π. vi is a vertex of at least three countries. (Since the cycle v1v2v3 is the
exterior cycle of the graph G, the vertex vi �= v1,v2,v3 is surrounded by countries
of G. Since G is 3-connected, vi cannot be a vertex of only one or two countries.)
Thus, for ϱ = 0, the curvature at vi is at most −π. Hence there is a unique number
ϱ>0 for which it vanishes. Now let ϱk+1,1 = ϱk+1,2 = ϱk+1,3 = 1,ϱk+1 j = ϱkj
for j = 4,...,n, j �= i and ϱk+1i = ϱ and go on to vertex vi+1.