# Goal Shot Analysis in Elite Water Polo--World Cup Final 2018 in Berlin
## Appendix A Direct Linear Transformation  
The goal of Direct Linear Transformation (DLT; e.g., [49; 50; 51] is to determine the actual location of the world coordinates \((x,y)^{T}\) based on the location of the images coordinates \((u,v)^{T}\). DLT transforms image coordinates of one (or more) cameras to the corresponding world coordinates of a point. Before this can be done, the system must be calibrated with points of known location (calibration points; for 2D-DLT at least six points; Figure A1). More points (for 2D-DLT more than six points) can enhance the estimate of \(\mathbf{L}\).  
\[\underbrace{\left(\begin{array}{ccccccccc}x_{1}&y_{1}&1&0&0&0&-u_{1}x_{1}& -u_{1}y_{1}\\ 0&0&0&x_{1}&y_{1}&1&-v_{1}x_{1}&-v_{1}y_{1}\\ x_{2}&y_{2}&1&0&0&0&-u_{2}x_{2}&-u_{2}y_{2}\\ 0&0&0&x_{2}&y_{2}&1&-v_{2}x_{2}&-v_{2}y_{2}\\ \vdots&&&&\vdots\\ x_{N}&y_{N}&1&0&0&0&-u_{N}x_{N}&-u_{N}y_{N}\\ 0&0&x_{N}&y_{N}&1&-v_{N}x_{N}&-v_{N}y_{N}\\ \end{array}\right)}_{\text{2N}\times 8}\underbrace{\left(\begin{array}{c}L_{1} \\ L_{2}\\ L_{3}\\ L_{4}\\ L_{5}\\ L_{6}\\ L_{7}\\ L_{8}\\ \end{array}\right)}_{\text{8}\times 1}\underbrace{\left(\begin{array}{c}u_{1} \\ v_{1}\\ u_{2}\\ v_{2}\\ \vdots\\ u_{N}\\ \end{array}\right)}_{\text{2N}\times 1} \tag{18}\]  
or  
\[\mathbf{AL=B} \tag{19}\]  
Then, \(\mathbf{L}\) can be estimates by calibration to:  
\[\mathbf{L}=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A}^{T}\mathbf{B} \tag{20}\]  
This results in  
\[\underbrace{\left(\begin{array}{cc}L_{1}-L_{7}u&L_{2}-L_{8}u\\ L_{4}-L_{7}v&L_{5}-L_{8}v\\ \end{array}\right)}_{\text{Q}}\cdot\begin{pmatrix}x\\ y\end{pmatrix}=\underbrace{\left(\begin{array}{c}u-L_{3}\\ v-L_{6}\\ \end{array}\right)}_{\text{q}} \tag{21}\]or in case of two cameras (with indices \(L\) and \(R\) for left and right camera, respectively)  
\[\underbrace{\begin{pmatrix}L_{1}-L_{7}u_{L}&L_{2}-L_{8}u_{L}\\ L_{4}-L_{7}v_{L}&L_{5}-L_{8}v_{L}\\ R_{1}-R_{7}u_{R}&R_{2}-R_{8}u_{R}\\ R_{4}-R_{7}v_{R}&R_{5}-R_{8}v_{R}\end{pmatrix}}_{\begin{pmatrix}x\\ y\end{pmatrix}}=\underbrace{\begin{pmatrix}u_{L}-L_{3}\\ v_{L}-L_{6}\\ u_{R}-R_{3}\\ v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}u_{R}-R_{3}\\ v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v _{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v _{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v _{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v _{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix} \begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}}_{\begin{pmatrix}\begin{array}{c}x \\ y\end{pmatrix}\end{pmatrix}}_{\begin{array}{c}x\\ y\end{pmatrix}}=\underbrace{\begin{pmatrix}u_{L}-L_{3}\\ v_{L}-L_{6}\\ u_{R}-R_{3}\\ v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix}v_{R}-R_{6}\end{pmatrix}}_{\begin{pmatrix} \begin{array}{c}x\\ y\end{pmatrix}}_{\begin{array}{c}y\end{array}}\end{pmatrix}_{\begin{array}{c} x\\ y\end{pmatrix}}_{\begin{array}{c}y\end{array}} \tag{10}\]  
Rearrange to the world coordinates \((x,y)^{T}\) is:  
\[\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}\mathbf{Q}^{T}\mathbf{Q}\end{pmatrix}^{-1} \mathbf{Q}^{T}\mathbf{q} \tag{11}\]