The motion of a cutter tool is modeled as a surface undergoing a sweep operation along another geometric entity. A numerically controlled machining verification method is developed based on a formulation for delineating the volume generated by the motion of a cutting tool on the workpiece (stock). Varieties and subvarieties that are subsets of some Eucledian space defined by the zeros of a finite number of analytic functions are computed and are characterized as closed form equations of surface patches of this volume. A topological space describing the swept volume will be built as a stratified manifold with corners. Singularities of the variety are loci of points where the Jacobian of the manifold has lower rank than maximal. It is shown that varieties appearing inside the manifold representing the removed material are due to a lower degree strata of the Jacobian. Some of the varieties are complicated (so as not to confuse with varieties in complex Cn) and will be shown to be reducible because of their parametrization and are addressed. Benefits of this method are evident in its ability to depict the manifold and to compute a value for the volume.