In many cases, cell function is intimately linked to cell shape control. We used 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. Using microfabrication 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 regulation of myosin II by curvature and control of curvature by myosin II 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.