We describe algorithms for deterministically constructing discrete ε-approximations for continuous point sets such as distributions or terrains. Furthermore, for certain families of subsets , such as those described by axis-aligned rectangles, we reduce the size of the ε-approximations by almost a square root from to . This is often the first step in transforming a continuous problem into a discrete one for which combinatorial techniques can be applied. We describe applications of this result in geospatial analysis, biosurveillance, and sensor networks.