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.

翻译：当前，对偶四元数算法因其优势已被应用于三维坐标变换问题中。三维坐标变换问题是大地测量学中的重要问题之一。除大地测量学外，该变换问题也出现在许多其他应用领域。尽管存在多种坐标变换方法（相似变换、仿射变换、投影变换等），但相似变换因其简单性而被广泛采用。非对称变换因其易于使用而常优先于对称坐标变换。然而，从误差理论的角度来看，对称变换应更为优选。本研究探讨了基于对偶四元数算法的对称相似性三维坐标变换主题，分析了问题求解中遇到的瓶颈及其解决方法。提出了一种基于对偶四元数的新迭代算法。该算法通过两种不同模型实现：带约束方程和不带约束方程。同时评估了这两种模型相互比较的优缺点。不仅确定了变换参数，还估算了变换参数的误差。详细推导了估计三维变换参数对称相似性的公式，并逐步呈现。由于对称变换是非对称变换的一般形式，通过对所开发的对称变换模型进行简单修改，亦可获得非对称变换结果。所提算法能够同时执行二维和三维的对称与非对称相似变换。对于二维变换，仅需将两个系统中的z和Z坐标设为零即可。