# General Total Least Squares Theory for Geodetic Coordinate Transformations
## 2 Adaptation of the Transformation Models to the Constrained/Unconstrained TLS Problem
### Adaptation of 2D Transformations to the Functional Model of the TLS Problem  
The 2D affine transformation with six unknown parameters for a single point is introduced as:  
\[\left[ \begin{array}{cc}\mathbf{x}_{t^{\prime}}&\mathbf{y}_{t}\end{array}  \right]^{T}\approx\left[ \begin{array}{cc}m_{1}\cos\alpha&-m_{2}\sin(\alpha+ \varepsilon)\\ m_{1}\sin\alpha&m_{2}\cos(\alpha+\varepsilon)\end{array} \right]\left[ \begin{array} []{cc}\mathbf{x}_{s},&\mathbf{y}_{s}\end{array} \right]^{T}+\left[ \begin{array} []{cc}\Delta\mathbf{x}&\Delta y\end{array} \right]^{T}. \tag{3}\]  
\(\left[ \begin{array}{cc}\mathbf{x}_{t},&\mathbf{y}_{t}\end{array} \right]\) are the coordinates of the target system, while \(\left[ \begin{array}{cc}\mathbf{x}_{s},&\mathbf{y}_{s}\end{array} \right]\) are the coordinates of the source system. \(m_{1}\) and \(m_{2}\) are the scale factors, \(\alpha+\varepsilon\) and \(\alpha\) are the rotation angles, and \(\left[ \begin{array}{cc}\Delta\mathbf{x}&\Delta y\end{array} \right]\) are the translations. The sign \(\approx\) is used as we omit the random errors in the equation for the sake of simple formulation. When all six parameters are replaced by \(\xi=\left[vec^{T}\Big{(}\Xi_{2\times 2}\Big{)},\left[ \begin{array}{cc}\xi_{ \Delta\mathbf{x}}&\xi_{\Delta y}\end{array} \right]\right]^{T}\), the above equation is rewritten as:  
\[\left[ \begin{array}{cc}\mathbf{x}_{t},&\mathbf{y}_{t}\end{array} \right]^{T} \approx\Xi_{2\times 2}\left[ \begin{array}{cc}\mathbf{x}_{s},&\mathbf{y}_{s} \end{array} \right]^{T}+\left[ \begin{array}{cc}\xi_{\Delta\mathbf{x}}&\xi_{ \Delta y}\end{array} \right]^{T} \tag{4}\]  
with transformation matrix \(\Xi_{2\times 2}=\left[ \begin{array}{cc}\xi_{11}&\xi_{12}\\ \xi_{21}&\xi_{22}\end{array} \right]\). \(vec^{T}\) is the row vectorization operator according to the row order, i.e., the nth row stacks after the (n-1)th row. Considering all control points, the unconstrained EIV model of the affine transformation in 2D can be given by:  
\[\left[ \begin{array}{cc}\mathbf{x}_{t}^{T}&\mathbf{y}_{t}^{T}\end{array}  \right]^{T}\approx\left[\mathbf{I}_{2}\otimes\left[ \begin{array}{cc}\mathbf{ x}_{s}&\mathbf{y}_{s}\end{array} \right],\mathbf{I}_{2}\otimes 1\right]\xi. \tag{5}\]  
Equation (5) explicitly defines the matrix \(\mathbf{A}\) and the vector \(\mathbf{y}\) in Eq (1) for all other transformations in 2D. The operator \(\otimes\) denotes the Kronecker Product. \(\left[ \begin{array}{cc}\mathbf{x}_{s}&\mathbf{y}_{s}\end{array} \right]\) and \(\left[ \begin{array}{cc}\mathbf{x}_{t}&\mathbf{y}_{t}\end{array} \right]\) are the source x and y coordinate vectors and the target x and y coordinate vectors, respectively, for all control points. The vector 1 denotes the vector of ones with length the control point number. The orthogonal, similarity, and rigid transformation models for the single point:  
\[\left[ \begin{array}{cc}\mathbf{x}_{t},&\mathbf{y}_{t}\end{array} \right]^{T} \approx\left[ \begin{array}{cc}m_{1}\cos\alpha&-m_{2}\sin\alpha\\ m_{1}\sin\alpha&m_{2}\cos\alpha\end{array} \right]\left[ \begin{array}{cc} \mathbf{x}_{s},&\mathbf{y}_{s}\end{array} \right]^{T}+\left[ \begin{array}{cc} \Delta\mathbf{x}&\Delta y\end{array} \right]^{T}, \tag{6}\]  
\[\left[ \begin{array}{cc}\mathbf{x}_{t^{\prime}}&\mathbf{y}_{t}\end{array}  \right]^{T}\approx\left[ \begin{array}{cc}m\cos\alpha&-m\sin\alpha\\ m\sin\alpha&m\cos\alpha\end{array} \right]\left[ \begin{array}{cc}\mathbf{x}_{s },&\mathbf{y}_{s}\end{array} \right]^{T}+\left[ \begin{array}{cc}\Delta \mathbf{x}&\Delta y\end{array} \right]^{T}, \tag{7}\]  
\[\left[ \begin{array}{cc}\mathbf{x}_{t},&\mathbf{y}_{t}\end{array} \right]^{T} \approx\left[ \begin{array}{cc}\cos\alpha&-\sin\alpha\\ \sin\alpha&\cos\alpha\end{array} \right]\left[ \begin{array}{cc}\mathbf{x}_{s },&\mathbf{y}_{s}\end{array} \right]^{T}+\left[ \begin{array}{cc}\Delta \mathbf{x}&\Delta y\end{array} \right]^{T}, \tag{8}\]  
are simplified versions of Equation (3).