Abstract: For the classic perspective-three-point ( P3P) problem，when the Z-axis coordinates of the three-dimensional control points are randomly distributed in a large range，there are still problems of poor numerical stability， degradation caused by increased image noise，and low computational efficiency． Therefore，a fast and stable algebraic solution method was proposed in this article． Firstly，when the proposed solution directly estimates the rotation and position of a calibrated camera from three 3D to 2D point correspondences，an intermediate coordinate frame is introduced between the world coordinate frame and the camera coordinate frame to reduce the number of unknown parameters，and the rotation matrix is normalized to simplify the calculation process and improve the calculation efficiency． Secondly，the midpoint between the two control points was chosen as the origin point of the intermediate coordinate frame，so as to improve the anti-noise performance of the P3P problem in degenerate configurations． Finally，the P3P problem was transformed into a biquadratic equation with one unknown parameter by using a Grbner basis，then a closed solution to the P3P problem was obtained． Experimental results show that the proposed algorithm can achieve better numerical stability and anti-noise performance compared with other three classic algorithm of the P3P problem.

Key words: perspective-three-point problem, computer vision, camera, error analysis