We present a simple algorithm to generate a topology-preserving, error-bounded approximation of the outer boundary of the volume swept by a polyhedron along a parametric trajectory. Our approach uses a volumetric method that generates an adaptive volumetric grid, computes signed distance on the grid points, and extracts an isosurface from the distance field. In order to guarantee geometric and topological bounds, we present a novel sampling and front propagation algorithm for adaptive grid generation. We highlight the performance of our algorithm on many complex benchmarks that arise in geometric and solid modeling, motion planning and CNC milling applications. To the best of our knowledge, this is the first practical algorithm that can generate swept volume approximations with geometric and topological guarantees on complex polyhedral models swept along any parametric trajectory.