This paper presents a new paradigm for 3-axis tool path generation based on an incomplete 2-manifold mesh model, namely, an inexact polyhedron. When geometric data is transferred from one system to another system and tessellated for tool path generation, the model does not have any topological data between meshes and facets. In contrast to the existing polyhedral machining approach, the proposed method generates tool paths from an incomplete 2-manifold mesh model. In order to generate gouge-free tool paths, CL-meshes are generated by offsetting boundary edges, boundary vertices, and facets. The CL-meshes are sliced by machining planes and the calculated intersections are sorted, trimmed, and linked. The grid method is used to reduce the computing time when range searching problems arise. The method is fully implemented and verified by machining an incomplete 2-manifold mesh model.