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We propose a novel online point-based 3D reconstruction method from a posed monocular RGB video. Our model maintains a global point cloud scene representation but allows points to adjust their 3D locations along the camera rays they were initially observed. When a new RGB image is inputted, the model adjusts the location of the existing points, expands the point cloud with newly observed points, and removes redundant points. These flexible updates are achieved through our novel ray-based 2D-3D matching technique. Our point-based representation does not require a pre-defined voxel size and can adapt to any resolution. A unified global representation also ensures consistency from different views. Results on the ScanNet dataset show that we improve over previous online methods and match the state-of-the-art performance with other types of approaches.