The availability of low cost sensors has led to an unprecedented growth in the volume of spatial data. However, the time required to evaluate even simple spatial queries over large data sets greatly hampers our ability to interactively explore these data sets and extract actionable insights. Graphics Processing Units~(GPUs) are increasingly being used to speedup spatial queries. However, existing GPU-based solutions have two important drawbacks: they are often tightly coupled to the specific query types they target, making it hard to adapt them for other queries; and since their design is based on CPU-based approaches, it can be difficult to effectively utilize all the benefits provided by the GPU. As a first step towards making GPU spatial query processing mainstream, we propose a new model that represents spatial data as geometric objects and define an algebra consisting of GPU-friendly composable operators that operate over these objects. We demonstrate the expressiveness of the proposed algebra by formulating standard spatial queries as algebraic expressions. We also present a proof-of-concept prototype that supports a subset of the operators and show that it is at least two orders of magnitude faster than a CPU-based implementation. This performance gain is obtained both using a discrete Nvidia mobile GPU and the less powerful integrated GPUs common in commodity laptops.