We give efficient data-oblivious algorithms for several fundamental geometric problems that are relevant to geographic information systems, including planar convex hulls and all-nearest neighbors. Our methods are "data-oblivious" in that they don't perform any data-dependent operations, with the exception of operations performed inside low-level blackbox circuits having a constant number of inputs and outputs. Thus, an adversary who observes the control flow of one of our algorithms, but who cannot see the inputs and outputs to the blackbox circuits, cannot learn anything about the input or output. This behavior makes our methods applicable to secure multiparty computation (SMC) protocols for geographic data used in location-based services. In SMC protocols, multiple parties wish to perform a computation on their combined data without revealing individual data to the other parties. For instance, our methods can be used to solve a problem posed by Du and Atallah, where Alice has a set, A, of m private points in the plane, Bob has another set, B, of n private points in the plane, and Alice and Bob want to jointly compute the convex hull of A u B without disclosing any more information than what can be derived from the answer. In particular, neither Alice nor Bob want to reveal any of their respective points that are in the interior of the convex hull of A u B.