Analytical algorithm of weighted 3D datum transformation using the constraint of orthonormal matrix

Based on the Lagrangian extremum law with the constraint that rotation matrix is an orthonormal matrix, the paper presents a new analytical algorithm of weighted 3D datum transformation. It is a stepwise algorithm. Firstly, the rotation matrix is computed using eigenvalue-eigenvector decomposition. Then, the scale parameter is computed with computed rotation matrix. Lastly, the translation parameters are computed with computed rotation matrix and scale parameter. The paper investigates the stability of the presented algorithm in the cases that the common points are distributed in 3D, 2D, and 1D spaces including the approximate 2D and 1D spaces, and gives the corresponding modified formula of rotation matrix. The comparison of the presented algorithm and classic Procrustes algorithm is investigated, and an improved Procrustes algorithm is presented since that the classic Procrustes algorithm may yield a reflection rather than a rotation in the cases that the common points are distributed in 2D space. A simulative numerical case and a practical case are illustrated.