## 3 Results
### Path Generation and Visualization  
Most of the demand for metal parts deposited on non-planar surfaces is reflected in the growth of overhanging structures on a given component. Therefore, CAD models of this type of structure were chosen as the input to the algorithm. The four models used for testing are shown in Figure 14. These were modeled as parts with overhanging structures grown on non-planar surfaces, with triangle meshes fitted to their surfaces.  
Since different models have different geometrical features and dimensions, different parameters were set for them. The layer thickness parameter was uniformly set to 1.0 mm to display the results. The offset parameters were labeled in detail to verify the effect of combining helical paths and contour offsets. As shown in Figure 15, several twisted rectangular sheets were suspended from the ring body, which had its offset parameter set to \(\pm\)1.0 mm. Positive values represent outward expansion, and negative values represent inward contraction.  
As shown in Figure 15a, the path was a continuous helical shape curve, and the short lines growing above it were tool orientations. Since the offset parameter is negative, the path shrinks inside the model. The only way to see the model's path is to make it more transparent. As shown in Figure 15b, the path was positive due to the offset parameter, and it hovered outside the model. As shown in Figure 16, turbine A with thicker blades was selected for testing and it had an offset parameter of \(\pm\)1.5 mm.  
As shown in Figure 16a, the path was similarly constricted inside the blade. The path hovered around the outside of the blade as shown in Figure 16b. The effect of the offset was also more pronounced due to the increased offset distance. In addition to the cylindrical surface, paths were also generated on the spherical surface. As shown in Figure 17, some overhanging cylinders were distributed on the surface of the sphere. The offset parameter of the radiation sphere was set to \(\pm\)2.0 mm.  
Figure 14: Part models used for testing. (**a**) Ring body. (**b**) Turbine A. (**c**) Radiation sphere. (**d**) Turbine B.  
As shown in Figure 17, the helical path was contracted inside the cylinder. As shown in Figure 17, the helical path surrounded the surface of the cylinder. The method remained correct on the sphere where the surface was more curved. Finally, a freer variation in hub surface curvature and a greater degree of blade twist in the overhang were chosen on a complex model. As shown in Figure 18, the path was generated on turbine B, which had an offset parameter of \(\pm\)2.5 mm.  
As shown in Figure 18, the test was performed on a blade grown on a free-form surface. Even though the inward contraction was more complex and less fault-tolerant compared to the expansion, the path still exhibited contraction according to the given parameter distance. As shown in Figure 18, the path hovered around the outside of the blade with more complex surface curvature, showing an outwardly expanding pattern.  
Figure 16: Test results on turbine A. (**a**) Path contracted inward by 1.5 mm. (**b**) Path expanding outward by 1.5 mm.  
Figure 15: Test results on the ring. (**a**) Path contracted inward by 1.0 mm. (**b**) Path expanding outward by 1.0 mm.