Schematic representation of the main disk and the inner ring (corresponding to the blue and green substructures in Fig. 15). The stars represent the center of the ring (in green), and the center of M31 (in black). In our modeling, we neglect the component of the offset parallel to the line of sight, and only consider the offset toward the east direction (here $-\overrightarrow{u_y}$) and north direction (here $\overrightarrow{u_z}$). The x axis represents the line of sight, and the yOz plane the plane of sky. The green dotted and dashed axis is parallel to the z axis. In each color, the dashed lines represent two perpendicular diameters of the substructures. One of the two coincides to the major axis of the projection. The plain lines show where the second diameter would be if the substructure was not inclined (i.e., fully in the plane of sky). The angular parameters of the modeling are (i_r, θ_r) and (i_md, θ_md); the inclination and position angles of the ring and of the main disk.