We propose a novel approach for estimating the relative pose between rolling shutter cameras using the intersections of line projections with a single scanline per image. This allows pose estimation without explicitly modeling camera motion. Alternatively, scanlines can be selected within a single image, enabling single-view relative pose estimation for scanlines of rolling shutter cameras. Our approach is designed as a foundational building block for rolling shutter structure-from-motion (SfM), where no motion model is required, and each scanline's pose can be computed independently. % We classify minimal solvers for this problem in both generic and specialized settings, including cases with parallel lines and known gravity direction, assuming known intrinsics and no lens distortion. Furthermore, we develop minimal solvers for the parallel-lines scenario, both with and without gravity priors, by leveraging connections between this problem and the estimation of 2D structure from 1D cameras. % Experiments on rolling shutter images from the Fastec dataset demonstrate the feasibility of our approach for initializing rolling shutter SfM, highlighting its potential for further development. % The code will be made publicly available.