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Free space is represented as a union of (possibly overlapping) generalized cones. An algorithm is presented which efficiently finds good collision-free paths for convex polygonal bodies through space littered with obstacle polygons. The paths are good in the sense that the distance of closest approach to an obstacle over the path is usually far from minimal over the class of topologically equivalent collision-free paths. The algorithm is based on characterizing the volume swept by a body as it is translated and rotated as a generalized cone, and determining under what conditions one generalized cone is a subset of another.