A common way of storing spatio-temporal information about mobile devices is in the form of a 3D (2D geography + time) trajectory. We argue that when cellular phones and Personal Digital Assistants become location-aware, the size of the spatio-temporal information generated may prohibit efficient processing. We propose to adopt a technique studied in computer graphics, namely line-simplification, as an approximation technique to solve this problem. Line simplification uses a distance function in producing the trajectory approximation. We postulate the desiderata for such a distance: it should be sound, namely the error of the answers to spatiotemporal queries must be bounded. We analyze several distances, and prove that some are sound in this sense for some types of queries, while others are not. Interestingly, not a single distance analyzed proves to be sound for all the common spatio-temporal queries, and therefore multi-distance line-simplification is introduced and analyzed. The...