Determining local tangents and projected gradients. Upper: full projected gradient curve outside and during the transit. Lower: part of the transit where the projected gradient changes significantly and rapidly. Note: the projected gradient is stable for relatively large |x| because the change in the local ring edge tangent is negligible with position (the left and right asymptote are 1/s as dictated by Eq. (20)). For small |x|, the projected gradient rises to some peak value (where the local tangent is perpendicular to motion), before quickly dropping to 0 (where the local tangent is parallel to motion), and then rising once more. In the lower panel, it appears that the local tangent rotates counterclockwise with time, which supports the sinusoidal definition of the projected gradient (Eq. (21)).