Camera motion estimation, such as relative pose estimation and absolute pose estimation, are fundamental problems in computer vision and robotics. To obtain the motion parameters, classical methods rely on studying the properties of the geometric matrices, e.g., rotation matrix, essential matrix, homography matrix. The well known five-point algorithm was successfully derived using the singular constraint and trace constraints on the essential matrix. However, finding all the algebraic constraints is not always trivial for some recent problems. In this paper, we propose a simple and general technique to find complete algebraic constraints so that we can derive efficient algorithms. We show that using the quaternion to formulate the rotation matrix we can eliminate any unknowns from the original equations and obtain constraints on the rest of the unknowns based on Gröbner basis. We demonstrate that this approach can be applied to almost all the camera motion estimation and show its improvement compared to the existing methods. Further more, based on this elimination technique, we exploit new constraints for the relative pose estimation with gravity prior, and derive a new globally optimal algorithm to this problem. We compare our algorithm with the state-of-the-art methods on both synthetic and real-world data, and show the benefits including accuracy and efficiency.