Geometric methods for the conceptual design of nonideal and azeotropic distillation systems have focused primarily on three-and four-component mixtures. A generalization of geometric methods that does not rely on visualization and gives a completely equation-based approach to determine the structure of distillation regions for homogeneous mixtures is described. This method is useful for mixtures with a large number of components, provided a vapor-liquid equilibrium model is available The approach is applied to assess feasibility and alternative separations in several examples, including a five-component mixture with nine azeotropes and a six-component mixture with seven azeotropes. These methods can be used to rapidly generate process alternatives, and the examples demonstrate that there can be a surprising number of these alternatives for the distillation of azeotropic mixtures.