This paper presents an investigation of the origin and evolution of the complex flow pattern on a hemisphere-cylinder at separated flow conditions. Three-dimensional numerical simulations have been performed for a range of Reynolds numbers and angles of attack. A critical point theory has been used to analyze the flowfields. This has yielded, for the first time for this geometry, a bifurcation diagram that classifies the different flow topology regimes as a function of the Reynolds number and angle of attack. A complete characterization of the origin and evolution of the complex structural patterns of this geometry is documented. For the higher Reynolds number and angle of attack, a structurally stable topology is found that is associated with the pattern of the horn vortices, usually found on this geometry in a range from low to high Reynolds numbers and from incompressible to compressible regimes. Surface critical points and surface and volume streamlines describe the main flow structures and their strong dependence with the flow conditions.