We study the problem of shortest-path geographic routing in a static sensor network. Existing algorithms often make routing decisions based on node information in local neighborhoods. However, it is shown by Kuhn et al. that such a design constraint results in a highly undesirable lower bound for routing performance: if a best route has length c, then in the worst case a route produced by any localized algorithm has length Ω(c^2), which can be arbitrarily worse than the optimal. We present VIGOR, a VIsibility-Graph-based rOuting pRotocol that produces routes of length Θ(c). Our design is based on the construction of a much reduced visibility graph, which guides nodes to find near-optimal paths. The per-node protocol overheads in terms of state information and message transmission depend only on the complexity of the field's large topological features, rather than on the network size. Simulation results show that our protocol dramatically outperforms localized protocols such as GPSR and GOAFR+ in both average and worst cases, with reasonable extra overheads.