The Discrete Schwarz-Pick Lemma For Circle Packings Revisited

The Discrete Schwarz-Pick Lemma is a discrete analogue of the classical result from complex analysis, arising from the connection between circle packings and conformal maps established by Thurston. Previous works by Beardon-Stephanson and Van Eeuwen proved this lemma for circle packings where circles are tangent or intersect at non-obtuse angles, corresponding to inversive distances $I \in [0,1]$. This paper extends the investigation to circle packings with obtuse intersections ($I \in (-1,0)$) and disjoint packings ($I>1$). We prove that the Discrete Schwarz-Pick Lemma holds for the full range of intersecting circle packings with inversive distances in $(-1,1]$, provided an additional condition on the weights of each triangle is satisfied. The proof relies on a variational principle for circle packings with inversive distances. Conversely, we show that the lemma fails for disjoint circle packings where $I\geq1$. This is demonstrated by constructing a specific counterexample on a triangulated disk with four vertices.