  We provide a simple characterization of the critical temperature for the
Ising model on an arbitrary planar doubly periodic weighted graph. More
precisely, the critical inverse temperature \beta for a graph G with coupling
constants (J_e)_{e\in E(G)} is obtained as the unique solution of a linear
equation in the variables (\tanh(\beta J_e))_{e\in E(G)}. This is achieved by
studying the high-temperature expansion of the model using Kac-Ward matrices.
