Comparative Study on the Suitability of Free-Boundary Parameterization Techniques for 3D Geometry-Aware Path Generation

This paper presents a comparative study on the suitability of free-boundary surface parameterization techniques for generating trajectories on 3D surfaces. The approach maps a 3D surface to a 2D parametric domain through four parameterization methods: Least-Squares Conformal Mapping, Boundary First Flattening, As-Rigid-As-Possible, and Conformal Equivalence of Triangular Meshes. Structured trajectory patterns are generated in the 2D domain and projected back to 3D. We introduce center-to-boundary geodesic deviation measure, which yields a deviation profile over the boundary loop and reflects how well central alignment is preserved under each parameterization method. The results highlight differences in distortion and geodesic preservation, reflecting the suitability of methods for path generation.