# A Fast and Simple Method for Absolute Orientation Estimation Using a Single Vanishing Point
## 2 Materials and Methods
### Limitation of Parallel Lines  
In Section 2.1, only one vanishing point is used to estimate the camera orientation and this expands the range of application for our proposed method. However, the limitation of parallel lines exists in this paper. In this section, we assume this limitation, i.e., \(d_{w}\
eq\left(\begin{matrix}0&0&1\end{matrix}\right)\) and its algebraic derivation is conducted as follows.  
Here, we assume the unit direction vector \(d_{w}\) of the parallel lines of \(L_{1}\) and \(L_{2}\) in the world frame is \(\left(\begin{matrix}0&0&1\end{matrix}\right)\). From Formulas (8) and (9), we can obtain  
\[\begin{bmatrix}d_{cx}\\ d_{cy}\\ d_{cz}\end{bmatrix}=R_{\text{w},c}\cdot\begin{bmatrix}d_{wx}\\ d_{wy}\\ d_{wz}\end{bmatrix}=R_{\text{w},c}\cdot\begin{bmatrix}0\\ 0\\ 1\end{bmatrix} \tag{17}\] \[R_{\text{w},c}=\begin{bmatrix}\cos a_{z}&\sin a_{z}&0\\ -\cos a_{x}\sin a_{z}&\cos a_{x}\cos a_{z}&\sin a_{x}\\ \sin a_{x}\sin a_{z}&-\sin a_{x}\cos a_{z}&\cos a_{x}\end{bmatrix}\]  
Consequently, the unit direction vector \(d_{c}\) in the camera frame can be given using  
\[d_{cx} =0 \tag{18}\] \[d_{cy} =\sin a_{x}\] \[d_{cz} =\cos a_{x}\]  
From the first equation of formula (8), a new equation can be given  
\[d_{cx}\cos a_{z}+\left(d_{cz}\sin a_{x}-d_{cy}\cos a_{x}\right)\sin a_{z}=0 \tag{19}\]  
Equation (19) is rewritten using formula (18), and a new equation is obtained  
\[0\cdot\cos a_{z}+0\cdot\sin a_{z}=0 \tag{20}\]  
From Equation (20), it can be seen that \(a_{z}\) can be an arbitrary value. This means there are infinite solutions of camera orientation and hence we cannot estimate the orientation if \(d_{w}=\left(\begin{matrix}0&0&1\end{matrix}\right)\).  
After analyzing the limitation of our proposed method theoretically, we analyzed it using intuitional geometry. The projections of parallel lines of \(L_{1}\) and \(L_{2}\) whose direction vector is \(\left(\begin{matrix}0&0&1\end{matrix}\right)\) in world frame are illustrated in Figure 2.  
Figure 2: The projections of parallel lines of \(L_{1}\) and \(L_{2}\), whose direction vector is a limitation of our proposed method.  
In Figure 2, the initial world frame \(O_{w1}-X_{w1}Y_{w1}Z_{w1}\) is rotated around the \(Z\)-axis by \(\alpha\) degrees to the new world frame \(O_{wn}-X_{wn}Y_{wn}Z_{wn}\). It can be seen that the direction vector of parallel lines of \(L_{1}\) and \(L_{2}\) is not changed, and is still \(\big{(}\begin{array}{cc}0&0&1\end{array}\big{)}\). Moreover, the spatial positions of the two lines in the camera frame \(O_{c}-X_{c}Y_{c}Z_{c}\) are also not changed, which means their image projections of \(l_{1}\) and \(l_{2}\) are not changed after rotating. In brief, the 2D-3D line correspondences of parallel lines \(L_{1}\) and \(L_{2}\) are not changed when the world frame is rotated around the \(Z\)-axis, and hence the corresponding vanishing point is unchanged. Now, this vanishing point and the direction vector can be obtained from the camera frame \(O_{c}-X_{c}Y_{c}Z_{c}\) and world frame \(O_{w1}-X_{w1}Y_{w1}Z_{w1}\), respectively, but can also be obtained from the camera frame \(O_{c}-X_{c}Y_{c}Z_{c}\) and world frame \(O_{wn}-X_{wn}Y_{wn}Z_{wn}\), respectively. The phenomenon shows that infinite solutions can be estimated using parallel lines whose direction vector is \(\big{(}\begin{array}{cc}0&0&1\end{array}\big{)}\), and we cannot obtain the uniqueness. This is consistent with the results of the algebraic derivation above.  
Hence, if \(d_{w}=\big{(}\begin{array}{cc}0&0&1\end{array}\big{)}\), we use another rotation order and the new rotation matrix is  
\[R_{w\_c2}=\left[\begin{array}{ccc}\cos a_{x}&\sin a_{x}\cos a_{z}&\sin a_{x} \sin a_{z}\\ -\sin a_{x}&\cos a_{x}\cos a_{z}&\sin a_{z}\cos a_{x}\\ 0&-\sin a_{z}&\cos a_{z}\end{array}\right] \tag{21}\]  
Now, we can obtain the solution following the steps in Section 2.1. Here, note that the direction vector cannot be \(\big{(}\begin{array}{cc}1&0&0\end{array}\big{)}\).