In five-axis machining, generating gouge-free tool paths efficiently is essential. However, the projection method – widely regarded as an effective approach for gouge-free machining – remains largely applied to fixed tool-axis tool-path computation due to practical difficulties. This paper presents a comprehensive framework for projecting a cutter along arbitrary direction onto triangulated models, focusing on resolving the critical torus-edge tangency problem. We demonstrate that for two specific cases, tool-axis-aligned torus-edge projection and arbitrary-direction ball-end cutter projection, the problem can be reduced to a quartic equation and solved analytically, ensuring both robustness and computational efficiency. For the general case of arbitrary-direction torus-edge projection, we develop three tailor-made numerical methods: Bézier clipping, edge-search, and torus-search. The proposed method is validated through two applications: gouge avoidance in blade machining and gouge-free iso-planar tool path generation. Simulations and physical machining experiments confirm the method’s practical effectiveness and computational performance, demonstrating its readiness for industrial use.
Li et al. (Fri,) studied this question.