# Uncertainty Analysis of 3D Line Reconstruction in a New Minimal Spatial Line Representation
## Appendix A Jacobian of Conversion from a Plucker Line to the Proposed Representation  
Given a Plucker line \(\mathbf{L}=(\mathbf{m};\mathbf{d})\), where \(\mathbf{m}=(m_{1},m_{2},m_{3})^{\mathrm{T}}\) and \(\mathbf{d}=(d_{1},d_{2},d_{3})^{\mathrm{T}}\), we can convert it into the proposed representation by \((\theta,\phi,m_{l},\alpha)\). The Jacobian of the conversion is derived:  
\[\frac{\partial\theta}{\partial\mathbf{L}}=\left[0\quad 0\quad 0\quad 0\quad 0 \quad-\frac{1}{1-d_{3}^{2}}\right], \tag{1}\]  
\[\frac{\partial\phi}{\partial\mathbf{L}}=\left[0\quad 0\quad 0\quad-\frac{d_{2}}{d_{1}^{2}+d_{2}^{ 2}}\quad\frac{d_{1}}{d_{1}^{2}+d_{2}^{2}}\quad 0\right], \tag{2}\]  
\[\begin{split}\frac{\partial m_{l}}{\partial\mathbf{L}}& =\left[0\quad 0\quad 0\quad\frac{m_{1}}{\sqrt{m_{1}^{2}+m_{2}^{2}+m_{3}^{2}}} \quad\frac{m_{2}}{\sqrt{m_{1}^{2}+m_{2}^{2}+m_{3}^{2}}}\quad\frac{m_{3}}{ \sqrt{m_{1}^{2}+m_{2}^{2}+m_{3}^{2}}}\right]\\ &=\left[0\quad 0\quad 0\quad\frac{m_{1}}{|\mathbf{m}|}\quad \frac{m_{2}}{|\mathbf{m}|}\quad\frac{m_{3}}{|\mathbf{m}|}\right].\end{split} \tag{3}\]  
As for \(\frac{\partial\alpha}{\partial\mathbf{L}}\), it is more complicated. In the following we give the partial derivatives of \(\alpha\) to \(\mathbf{m}\) and \(\mathbf{d}\), respectively.  
When \(\mathbf{d}^{\mathrm{T}}(\mathbf{v}_{s}\times(\mathbf{d}\times\mathbf{m}))\geq 0\),  
\[\begin{split}\frac{\partial\alpha}{\partial\mathbf{m}}& =-\frac{1}{\sqrt{1-(\frac{\mathbf{v}_{s}^{\mathrm{T}}(\mathbf{d} \times\mathbf{m})}{|\mathbf{m}|})^{2}}}\cdot\frac{\partial\frac{\mathbf{v}_{s}^ {\mathrm{T}}(\mathbf{d}\times\mathbf{m})}{|\mathbf{m}|}}{\partial\mathbf{m}}\\ &=-\frac{1}{\sqrt{\sin^{2}(\alpha)}}\cdot\frac{|\mathbf{m}|\, \mathbf{v}_{s}^{\mathrm{T}}\,[\mathbf{d}]_{\times}+\frac{\partial\mathbf{m}}{ |\mathbf{m}|}\cdot((\mathbf{d}\times\mathbf{m})^{\mathrm{T}}\mathbf{v}_{s})}{ |\mathbf{m}|^{2}}\\ &=-\frac{1}{|\sin(\alpha)|}\cdot\frac{|\mathbf{m}|\,\mathbf{v}_{s }^{\mathrm{T}}\,[\mathbf{d}]_{\times}+\mathbf{m}\cos(\alpha)}{|\mathbf{m}|^{2} },\end{split} \tag{4}\]\[\begin{split}\frac{\partial\alpha}{\partial\mathbf{d}}& =-\frac{1}{\sqrt{1-(\frac{\mathbf{v}_{s}^{\mathrm{T}}(\mathbf{d}\times \mathbf{m})}{|\mathbf{m}|})^{2}}}\cdot\frac{1}{|\mathbf{m}|}\cdot\frac{ \partial\mathbf{v}_{s}^{\mathrm{T}}(\mathbf{d}\times\mathbf{m})}{\partial \mathbf{d}}\\ &=-\frac{1}{|\sin(\alpha)|\cdot|\mathbf{m}|}((\mathbf{d}\times \mathbf{m})^{\mathrm{T}}\cdot\frac{\partial\mathbf{v}_{s}}{\partial\mathbf{d} }+\mathbf{v}_{s}^{\mathrm{T}}\cdot\frac{\partial(\mathbf{d}\times\mathbf{m})}{ \partial\mathbf{d}})\\ &=-\frac{1}{|\sin(\alpha)|\cdot|\mathbf{m}|}((\mathbf{d}\times \mathbf{m})^{\mathrm{T}}\cdot\frac{\partial\mathbf{v}_{s}}{\partial\mathbf{d} }-\mathbf{v}_{s}^{\mathrm{T}}\cdot[\mathbf{m}]_{\times}),\end{split} \tag{10}\]  
where\(\frac{\partial\mathbf{v}_{s}}{\partial\mathbf{d}}=\frac{\partial\mathbf{v}_{s}}{ \partial(\theta,\phi)}\cdot\frac{\partial(\theta,\phi)}{\partial\mathbf{d}}\) and  
\[\frac{\partial\mathbf{v}_{s}}{\partial(\theta,\phi)}=\begin{bmatrix}-\sin \theta\cos\phi&-\cos\theta\sin\phi\\ -\sin\theta\sin\phi&\cos\theta\cos\phi\\ -\cos\theta&0\end{bmatrix}, \tag{11}\]  
\[\frac{\partial(\theta,\phi)}{\partial\mathbf{d}}=\begin{bmatrix}0&0&-\frac{1 }{\sqrt{1-d_{s}^{3}}}\\ \frac{d_{s}}{d_{1}^{2}+d_{2}^{2}}&\frac{d_{1}}{d_{1}^{2}+d_{2}^{2}}&0\end{bmatrix}. \tag{12}\]  
When \(\mathbf{d}^{\mathrm{T}}(\mathbf{v}_{s}\times(\mathbf{d}\times\mathbf{m}))<0\), the partial derivative \(\frac{\partial\mathbf{v}}{\partial\mathbf{L}}\) has a negative sign from the one above.