arXiv:2610.08486v1 Announce Type: new Abstract: Path following requires an agent to converge to and traverse a geometric path without a prescribed timing law. Guiding vector fields (GVFs) address this directly, but as guidance laws they offer no mechanism to compensate for the agent's own dynamics. Differential flatness-based control (DFBC), on the other hand, compensates for the agent's nonlinear dynamics but relies on a time-parameterized reference trajectory. This letter presents a unified algorithm, which we term flatness-based geometric control (FGC), that generates the DFBC reference hierarchy directly from a GVF and its higher-order time derivatives, eliminating the need for a pre-planned time-parameterized trajectory while retaining full dynamics-informed control. A cascade Lyapunov argument establishes exponential convergence of the tracking error and the physical trajectory to the desired path. The framework is further extended to singularity-free GVFs, which removes the topological obstruction and guarantees the condition for global exponential convergence on closed or self-intersecting paths. The proposed algorithm is validated in simulation.