A wide range of man-made environments can be abstracted as the Atlanta world. It consists of a set of Atlanta frames with a common vertical (gravitational) axis and multiple horizontal axes orthogonal to this vertical axis. This paper focuses on leveraging the regularity of Atlanta world for monocular SLAM. First, we robustly cluster image lines. Based on these clusters, we compute the local Atlanta frames in the camera frame by solving polynomial equations. Our method provides the global optimum and satisfies inherent geometric constraints. Second, we define the posterior probabilities to refine the initial clusters and Atlanta frames alternately by the maximum a posteriori estimation. Third, based on multiple local Atlanta frames, we compute the global Atlanta frames in the world frame using Kalman filtering. We optimize rotations by the global alignment and then refine translations and 3D line-based map under the directional constraints. Experiments on both synthesized and real data have demonstrated that our approach outperforms state-of-the-art methods.