Gruber P. Convex and Discrete Geometry

34 Koebe’s Representation Theorem for Planar Graphs 509
ϑv(σ ) =···+arctan ρw
+···
λρv
=···+arctan ρw
�
1
� �
1
�
+···+const − 1 + o − 1
ρv
λ λ
= ϑv(ρ) + const(1 − λ) + o(1 − λ) as λ → 1 − 0,
where const denotes a positive constant. Similarly,
(21) ϑv(ρ) ≥ ϑv(σ ) for each v �∈ T
and an application of Taylor’s theorem shows that the following hold:
(22) For each v �∈ T with ϑv(ρ) = 0, ϑv(σ ) = ϑv(ρ) + o(1 − λ) as λ →
1 − 0.
For all λ sufficiently close to 1, we have σ ∈ R (see the definition of R) and Propositions
(19)–(22) show that
µ(ρ) − µ(σ) = �
v∈V(G ′ )
ϑv(ρ) 2 − ϑv(σ ) 2
= � �
ϑv(ρ) + ϑv(σ ) �� ϑv(ρ) − ϑv(σ ) � − �
v∈T
+ �
v�∈T
ϑv (ρ)>0
v�∈T
ϑv (ρ)=0
� ϑv(ρ) + ϑv(σ ) �� ϑv(ρ) − ϑv(σ ) � > 0.
ϑv(σ ) 2
This contradicts the minimality of µ(ρ), thus concluding the proof of (18) which, in
turn, implies (9).
Having proved (1), (5) and (9), the proof of the theorem is complete. ⊓⊔
34.2 Thurston’s Algorithm and the Riemann Mapping Theorem
In view of the numerous applications of the Koebe-Andreev-Thurston-Bright-well-
Scheinerman theorem, the problem arises to construct the disc packings in an
effective way. The first procedure to serve this purpose for graphs with triangular
countries was proposed by Thurston [1000]. It is called Thurston’s algorithm,
although it is not an algorithm in the strict sense which is always finite. Rodin and
Sullivan [844] and Colin de Verdière [214, 215] showed its convergence. See also
Collins and Stephenson [216]. Mohar [747] considered the more general case of
primal-dual circle packings and gave a polynomial time algorithm.
In the following we first describe the algorithm of Thurston, following Rodin
and Sullivan. Then the relation between packing of discs and the Riemann mapping
theorem will be described. No proofs are given. For the figures we are indebted to
Kenneth Stephenson [968].
For more information, see the references given in the introduction of Sect. 34.