We present a method for calculating the envelope surface of a parametric solid object swept along a path in three-dimensional space. The boundary surface of the solid is the combination of parametric surfaces and an implicit surface where the Jacobian of the defining function has a rank-deficiency condition. Using this condition, we determine a set of square sub-Jacobian determinants that must all vanish simultaneously on the implicit surface. When the generator of the swept surface is a trivariate tensor-product B-spline solid and the path is a B-spline curve, we can give a robust algorithm to determine the implicit surface. This algorithm is based upon the “marching tetrahedra” method, which is adapted to work on 4-simplices. The union of the parametric and implicit surfaces gives the envelope of the swept solid.