Consistency of 2D and 3D distances of intuitionistic fuzzy sets

For intuitionistic fuzzy sets, it is argued that three dimensional distance functions are not necessary because two dimensional distance functions already provide a simple and concise expression of the distance between two intuitionistic fuzzy sets. However, we show that a three dimensional interpretation of intuitionistic fuzzy sets could give different comparison results to the ones obtained with their two dimensional counterparts. In addition to the existing distances, we define a three dimensional Hausdorff distance and compare its consistency with its two dimensional counterpart, which shows the usefulness of the three dimensional functions to model and provide the expression of the distance between two intuitionistic fuzzy sets.