Computing the boundary surface of the 3D volume swept by a rigid or deforming solid remains a challenging problem in geometric modeling. Existing approaches are often limited to sweeping rigid shapes, cannot guarantee a watertight surface, or struggle with modeling the intricate geometric features (e.g., sharp creases and narrow gaps) and topological features (e.g., interior voids). We make the observation that the sweep boundary is a subset of the projection of the intersection of two implicit surfaces in a higher dimension, and we derive a characterization of the subset using winding numbers. These insights lead to a general algorithm for any sweep represented as a smooth time-varying implicit function satisfying a genericity assumption, and it produces a watertight and intersection-free surface that better approximates the geometric and topological features than existing methods.