Azeotropy is a thermodynamic phenomenon where liquid and vapour coexisting phases have the same composition. In binary mixtures, the azeotropy calculation is represented by a 2x2 nonlinear system of algebraic equations with temperature (or pressure) and one molar fraction as unknowns. On rare occasions, this nonlinear system exhibits two solutions, characterizing a double azeotrope. In this work, we calculate double azeotropes with a geometry-based methodology: the numerical inversion of functions from the plane to the plane. We present results for two mixtures where the double azeotropy phenomenon occurs: the system benzene+hexafluorobenzene and the system 1,1,1,2,3,4,4,5,5,5-decafluoropentane+oxolane. The persistence of double azeotropes across different pressures is made clear by this global geometric approach. Moreover the vanishing of the existence of a pair of azeotropes is explained by the coalescence of them in just one, as can be easily understood from a global geometric viewpoint presented of the nonlinear function involved. The results indicate that this methodology can be a powerful tool for a better understanding of nonlinear algebraic systems in phase coexistence problems.