Symmetric similarity 3D coordinate transformation based on dual quaternion algorithm

Nowadays, we have seen that dual quaternion algorithms are used in 3D coordinate transformation problems due to their advantages. 3D coordinate transformation problem is one of the important problems in geodesy. This transformation problem is encountered in many application areas other than geodesy. Although there are many coordinate transformation methods (similarity, affine, projective, etc.), similarity transformation is used because of its simplicity. The asymmetric transformation is preferred to the symmetric coordinate transformation because of its ease of use. In terms of error theory, the symmetric transformation should be preferred. In this study, the topic of symmetric similarity 3D coordinate transformation based on the dual quaternion algorithm is discussed, and the bottlenecks encountered in solving the problem and the solution method are discussed. A new iterative algorithm based on the dual quaternion is presented. The solution is implemented in two different models: with constraint equations and without constraint equations. The advantages and disadvantages of the two models compared to each other are also evaluated. Not only the transformation parameters but also the errors of the transformation parameters are determined. The detailed derivation of the formulas for estimating the symmetric similarity of 3D transformation parameters is presented step by step. Since symmetric transformation is the general form of asymmetric transformation, we can also obtain asymmetric transformation results with a simple modification of the model we developed for symmetric transformation. The proposed algorithm is capable of performing both 2D and 3D symmetric and asymmetric similarity transformations. For the 2D transformation, it is sufficient to replace the z and Z coordinates in both systems with zero.