In this work, we propose a computational strategy and methods for the automated calculation of complete loci of homogeneous azeotropy of binary mixtures and the related Pxy and Txy diagrams for models of the equation-of-state (EOS) type. The strategy consists of first finding the system's azeotropic end points (AEPs). These can exist on vapor-liquid (VL) critical lines (CAEPs), on liquid-liquid-vapor (LLV) lines (HAEPs), and on pure-compound vapor pressure lines (PAEPs). Next, for the chosen binary system, we generate one or two azeotropic lines. Each of these lines has, as its starting point, one of the previously identified AEPs. We calculate the azeotropic lines using a numerical continuation method that solves the nonlinear azeotropic system of equations under a range of conditions and efficiently tracks entire azeotropic curves. We have integrated our strategy for calculating azeotropic lines into a general algorithm for the single-run computation of binary global phase equilibrium diagrams (GPEDs). GPEDs are defined by pure-compound, critical, LLV, and azeotropic lines. We implemented this general algorithm in the computer program GPEC (Global Phase Equilibrium Calculations), which makes it possible to evaluate, at a glance, the behavior of a given model-parameter values combination, for a chosen model and binary system.