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This paper introduces a novel antipodal-epipolar constraint on relative camera motion. By using antipodal points, which are available in large Field-of-View cameras, the translational and rotational motions of a camera are geometrically decoupled, allowing them to be separately estimated as two problems in smaller dimensions. We present a new formulation based on discrete camera motions, which works over a larger range of motions compared to previous differential techniques using antipodal points. The use of our constraints is demonstrated with two robust and practical algorithms, one based on RANSAC and the other based on Hough-like voting. As an application of the motion decoupling property, we also present a new structure-from-motion algorithm that does not require explicitly estimating rotation (it uses only the translation found with our methods). Finally, experiments involving simulations and real image sequences will demonstrate that our algorithms perform accurately and robustly, with some advantages over the state-of-the-art.