This paper outlines a novel, comprehensive framework for geolocalization, that is, determining the physical location of Internet hosts based on network measurements. The core 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 using cheap and accurate geometric methods. The framework is general and accommodates both positive and negative constraints, that is, constraints on where the node can or cannot be, respectively. It can reason in the presence of uncertainty, enabling it to gracefully cope with aggressively derived constraints that may contain errors. Since the solution space is represented geometrically as a region bounded by Bezier curves, the framework yields an accurate set of all points where the target may be located. Preliminary results on PlanetLab show promise; the framework can localize the median node to within 22 mi., a factor of three better than previous approaches, with little error.