Angiogenesis is the process by which new blood vessels develop from existing vasculature. During angiogenesis, endothelial tip cells migrate via diffusion and chemotaxis, loops form via tip-to-tip and tip-to-sprout anastomosis, new tip cells are produced via branching, and a vessel network forms as endothelial cells follow the paths of tip cells. The latter process is known as the snail trail. We use a mean-field approximation to systematically derive a continuum model from a two-dimensional lattice-based cellular automaton model of angiogenesis in the corneal assay, based on the snail-trail process. From the two-dimensional continuum model, we derive a one-dimensional model which represents angiogenesis in two dimensions. By comparing the discrete and one-dimensional continuum models, we determine how individual cell behavior manifests at the macroscale. In contrast to the phenomenological continuum models in the literature, we find that endothelial cell creation due to tip cell movement (vessel formation via the snail trail) manifests as a source term of tip cells on the macroscale. Further, we find that phenomenological continuum models, which assume that endothelial cell creation is proportional to the flux of tip cells in the direction of increasing chemoattractant concentration, qualitatively capture vessel formation in two dimensions, but must be modified to accurately represent vessel formation. Additionally, we find that anastomosis imposes restrictions on cell density, which, if violated, leads to ill-posedness in our continuum model. We also deduce that self-loops should be excluded when tip-to-sprout anastomosis is active in the discrete model to ensure propagation of the vascular front.