In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS 3 ). If the Wilson loop is given by a boundary curve $ \overrightarrow{X} $ ( s ) we define, using the integrable properties of the system, a family of curves $ \overrightarrow{X} $ ( λ, s ) depending on a complex parameter λ known as the spectral parameter. This family has remarkable properties. As a function of λ , $ \overrightarrow{X} $ ( λ, s ) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, $ \overrightarrow{X} $ ( λ, s ) has an essential singularity at the origin λ = 0. The coefficients of the expansion of $ \overrightarrow{X} $ ( λ, s ) around λ = 0, when appropriately integrated along the curve give the area of the corresponding minimal area surface.