# General Total Least Squares Theory for Geodetic Coordinate Transformations
## 6 Conclusions and Outlook  
In this contribution, we presented transformation models in the context of the TLS method, developed the corresponding algorithm based on constrained nonlinear normal equations, and provided a statistical assessment of the TLS adjustment results, including the cofactor matrix of the parameter estimator and the a posteriori variance factor.  
In the adaptation of the transformation problems to the EIV model, we explicitly expressed the functional and stochastic models by the source and target coordinates and emphasized the differences between the transformations distinguished only by the quadratic constraints. The adaptations to 2D and 3D are quite similar. In particular, the structure of the matrix \(A\) and the vector \(\mathbf{y}\) need to be enlarged by the z coordinates in the assigned place. For the adaptation, it is important to note that the number of constraints equals the number of the transformation matrix minus the number of independent parameter numbers. The number of constraints fixes the degree of freedom in the model.  
After formulating the transformation models using an unconstrained or constrained EIV model, Lagrange multipliers were applied to provide the first-order necessary conditions of the TLS optimization. After some rearrangements, the constrained nonlinear normal equations were established, based on which the Newton-type iterative procedure could be implemented. The further advantage of the formulation of the constrained nonlinear normal equations is that one can explicitly compute the cofactor matrix and the variance factor, unlike with other existing methods, e.g., the sequential quadratic program.  
We applied the proposed algorithm to selected examples so as to present and explain the adjustment results of all transformations with regard to the objective function value and statistical characteristics. We showed that with more available constraints, the objective function values are larger and the cofactors of the parameter estimates are smaller. The numerical results correspond to the theoretical inference.  
This algorithm is not only valid for the case of many geodetic datum conversion problems, but also for other applications (photogrammetry, GIS, etc.) where the scale changes may be different or be fixed to one, which will justify the use of suitable constraints within the EIV model.  
Furthermore, we hope that the discussed model and the developed algorithm contribute to convincing many--not only geodetic--researchers that the benefits arising from the use of orthogonal regression analysis outweigh the additional effort. From the methodological point of view, our TLS estimation can be generalized to any M- or L-type estimation, which will be promising in robustifying data processing for large data sets such as point clouds.  
Y.Q.: Methodology and Edit; X.F.: Supervision and Idea; W.Z.: Methodology and Funding support; B.W.: Coding. All authors have read and agreed to the published version of the manuscript.  
We greatly thank three anonymous reviewers for their valuable comments. This research was supported by the National Natural Science Foundation of China (41774009, 41674002) and the Natural Science Foundation of Jiangsu Province (BK20180720).  
The authors declare no conflicts of interest