In many cases cell function is intimately linked to cell shape control. We utilized endothelial cell branching morphogenesis as a model to understand the role of myosin-II in shape control of invasive cells migrating in 3D collagen gels. We applied principles of differential geometry and mathematical morphology to 3D image sets to parameterize cell branch structure and local cell surface curvature. We find that Rho/ROCK-stimulated myosin-II contractility minimizes cell-scale branching by recognizing and minimizing local cell surface curvature. Utilizing micro-fabrication to constrain cell shape identifies a positive feedback mechanism in which low curvature stabilizes myosin-II cortical association, where it acts to maintain minimal curvature. The feedback between myosin-II regulation by and control of curvature drives cycles of localized cortical myosin-II assembly and disassembly. These cycles in turn mediate alternating phases of directionally biased branch initiation and retraction to guide 3D cell migration.