Hexapod-type external fixators based on a general 6-6 Stewart platform structure are extensively used to manage orthopaedic disorders. While implementing these robotic devices, a practical visual aid is needed to quickly identify their uncontrollable states referred to as singularities. Thus, a visible correlation between the singularity of hexapod-type external fixators and their particular configurations has been explored geometrically in this work. A novel method called stereographic projection is utilized for that purpose. A mathematical procedure has been established to determine the characteristic values of the singular states. It is found that in case four- out of six-rod directions intersect each other at a common point, two different singular robot configurations result. Besides, if four joint angles at the top and bottom rings of the hexapod are equal, rings being parallel, then five-rod directions are intersected by two lines each passing through the end joints of the fifth rod.