The first paper of this series [Tapp, M.; Holland, S. T.; Hildebrandt, D.; Glasser, D. Ind. Eng. Chem. Res. 2004,43 (2), 364] analyzed the behavior of ternary homogeneous distillation systems by using column profile maps, which are generated by using the difference point equation. Ideal thermodynamics was assumed, and it has been shown how using column profile maps one can devise more creative designs. This paper explores the mathematics and the topology of the maps for systems with ideal and nonideal thermodynamics by looking at the occurrence of singular points in ternary homogeneous systems depending on the values of the design parameters of the difference point and the reflux ratio. Eigenvalue and eigenvector maps have been introduced, and their usefulness for manipulating phase diagrams and therefore the column profiles has been shown. The focus of the third part lies in the application of the tools presented in this paper for distillation process synthesis and design.