During clathrin-mediated endocytosis, a patch of flat plasma membrane is internalized to form a vesicle. In mammalian cells, how the clathrin coat deforms the membrane into a vesicle remains unclear and two main hypotheses have been debated. The ''constant area'' hypothesis assumes that clathrin molecules initially form a flat lattice on the membrane and deform the membrane by changing its intrinsic curvature while keeping the coating area constant. The alternative ''constant curvature'' hypothesis assumes that the intrinsic curvature of the clathrin lattice remains constant during the formation of a vesicle while the surface area it covers increases. Previous experimental studies were unable to unambiguously determine which hypothesis is correct. In this paper, we show that these two hypotheses are only two extreme cases of a continuum spectrum if we account for the free energies associated with clathrin assembly and curvature generation. By tracing the negative gradient of the free energy, we define vesiculation pathways in the phase space of the coating area and the intrinsic curvature of clathrin coat. Our results show that, overall, the differences in measurable membrane morphology between the different models are not as big as expected, and the main differences are most salient at the early stage of endocytosis. Furthermore, the best fitting pathway to experimental data is not compatible with the constant-curvature model and resembles a constant-area-like pathway where the coating area initially expands with minor changes in the intrinsic curvature, later followed by a dramatic increase in the intrinsic curvature and minor change in the coating area. Our results also suggest that experimental measurement of the tip radius and the projected area of the clathrin coat will be the key to distinguish between models.