Trajectory collection is fundamental for location-based services but often involves sensitive information, such as a user's daily routine, raising privacy concerns. Local differential privacy (LDP) provides provable privacy guarantees for users, even when the data collector is untrusted. Existing trajectory collection methods ensure LDP only for discrete location spaces, where the number of locations affects their privacy guarantees and trajectory utility. Moreover, the location space is often naturally continuous, such as in flying and sailing trajectories, making these methods unsuitable. This paper proposes two trajectory collection methods that ensure LDP for continuous spaces: TraCS-D, which perturbs the direction and distance of locations, and TraCS-C, which perturbs the Cartesian coordinates of locations. Both methods are theoretically and experimentally analyzed for trajectory utility. TraCS can also be applied to discrete spaces by rounding perturbed locations to the nearest discrete points. It is independent of the number of locations and has only $\Theta(1)$ time complexity in each perturbation generation. Evaluation results on discrete location spaces validate this advantage and show that TraCS outperforms state-of-the-art methods with improved trajectory utility, especially for large privacy parameters.

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