Laser processing of freeform surfaces: Equidistant trajectory generation based on ARAP parameterization

Freeform surface laser processing has attracted growing attention for applications such as surface texturing, laser polishing, and microstructure fabrication. However, maintaining uniform spacing between adjacent scanning trajectories remains challenging because curvature-induced mapping distortion can lead to significant spacing inconsistency, particularly in regions with strong curvature variation. In this study, an equidistant trajectory generation method based on As-Rigid-As-Possible (ARAP) parameterization and explicit spacing control is proposed for freeform surface processing. The freeform surface is first mapped onto a low-distortion parameter domain for pattern localization, boundary reconstruction, and trajectory initialization. The initialized trajectory points are then mapped onto the three-dimensional surface, where an iterative spacing correction scheme is applied to explicitly enforce the equidistant constraint in physical space, and equidistant trajectories are finally constructed using a slicing-based geometric approach. To address the limitations of global parameterization under strong curvature variation, a parameter-domain partitioning strategy based on local surface geometry and pattern complexity is further introduced to localize geometric distortion and improve spacing consistency under complex surface conditions. Experimental results show that the proposed method improves trajectory spacing stability compared with conventional approaches, reducing the spacing deviation in high-curvature regions from 0.022 mm to approximately 0.001 mm. The improved spacing consistency also contributes to more uniform laser-material interaction and machining results. These results indicate that the proposed approach provides a practical method for equidistant trajectory generation on freeform surfaces and is applicable to precision laser surface processing.