We propose an original approach to absolute pose (AP) and Structure-from-Motion (SfM) which handles Rolling Shutter (RS) effects. Unlike most existing methods which either augment global shutter (GS) projection with velocity parameters or impose continuous time and motion through pose interpolation, we use local differential constraints. These are established by drawing analogies with non-rigid 3D vision techniques, namely Shape-from-Template (SfT) and Non-Rigid SfM (NRSfM). The proposed idea is to interpret the images of a rigid surface acquired by a moving RS camera as those of a virtually deformed surface taken by a GS camera. These virtually deformed surfaces are first recovered by relaxing the RS constraint using SfT or NRSfM. Then we upgrade the virtually deformed surface to the actual rigid structure and compute the camera pose and ego-motion by reintroducing the RS constraint. This uses a new 3D-3D registration procedure that minimizes a cost function based on the Euclidean 3D point distance. This is more stable and physically meaningful than the reprojection error or the algebraic distance used in previous work. Experimental results obtained with synthetic and real data show that the proposed methods outper