Determining the physical location of Internet hosts is a critical enabler for many new location-aware services. In this paper, we present Octant, a novel, comprehensive framework for determining the location of Internet hosts in the real world based solely on network measurements. The key insight behind this framework is to pose the geolocalization problem formally as one of error-minimizing constraint satisfaction, to create a system of constraints by deriving them aggressively from network measurements, and to solve the system geometrically to yield the estimated region in which the target resides. This approach gains its accuracy and precision by taking advantage of both positive and negative constraints, that is, constraints on where the node can and cannot be, respectively. The constraints are represented using regions bounded by B´ezier curves, allowing precise constraint representation and low-cost geometric operations. The framework can reason in the presence of uncertainty, ...