Mathematical Modeling and G-Code Generation for CNC Plasma Tube Notching at Arbitrary Intersection Angles

The tube-notching process is widely used to manufacture structural joints and ducting systems for fluid transport. In these applications, accurate intersection angles and proper fit-up geometry are essential to ensure reliable assembly and system performance. Consequently, CNC-based automation is increasingly adopted to improve productivity in operations where precision and cycle time are critical. The main problem, however, lies in the complexity of generating accurate cutting trajectories for tube–tube intersections and converting them into machine-executable commands. This study addresses this gap by proposing a simple, novel mathematical model for toolpath generation capable of producing intersection profiles at arbitrary joint angles, including lateral offset (non-coaxial) configurations. A systematic procedure was developed to convert the resulting trajectories into G-code, which was processed in a low-cost CNC plasma cutter designed to experimentally validate the toolpaths. The machine incorporates a fourth axis to enable bevel cutting during tube processing. Experimental results demonstrate stable operation, high dimensional accuracy (error ±0.1°), and consistent cut quality for trajectories generated by the proposed model, confirming the feasibility of the low-cost CNC plasma system and its scalability to diverse fabrication requirements.