In this paper, the analytic 3-RPR platform studied by Wenger and Chablat in 2009 will be analyzed regarding its cuspidality condition. Many investigations have paid great attention to the cuspidality phenomena that arise for some designs of the 3-RPR parallel manipulator. Nevertheless, most of them focus on obtaining the cusp points of direct kinematic singularity curves on a section of the joint space, meaning that one of the input variables must remain constant. The authors, in this paper, obtain the locus of cusp points of the manipulator under study in a complete analytic way and in 3-D joint space. This way, the three inputs can be actuated to perform a nonsingular transition that encircles one of the curves that belong to the locus of cusp points. It will be shown that the duplicity of one of the output variables of this manipulator causes the overlapping of two different cusp points in the joint space. In order to visualize this characteristic, initially, transitions in a section of the joint space will be analyzed using, additionally, the reduced configuration space. Then, transitions in the 3-D joint space will be performed, showing that, in order to ensure a feasible nonsingular transition, it is necessary to represent the direct kinematic singularity surface and assess the evolution of the three output variables along the trajectory.