## 2 Principles
### Light Plane Calibration Based on a Single Circle Target
#### 2.2.2 Single Circle Position and Posture Measurement  
The position and orientation of a circle are determined by the circle center and normal vector of the plane containing the circle, respectively. In this section, it is assumed that the intrinsic parameters of the camera are known. In Figure 4, \(O_{c}\) is the optical center of the camera, (\(O_{c}\); \(X_{c}\), \(Y_{c}\), \(Z_{c}\)) represents the CCS matching the WCS, (\(O_{uv}\); \(u\), \(v\)) is the CPCS, and \(R\) denotes the radius of circle \(C\). Using the image processing algorithm described in Section 2.2.1, the elliptic fitting profile \(E\) of the circle target edge can be obtained. The main objectives of the process are (1) solving the elliptic cone equation of \(Q\), (2) converting \(Q\) from the CCS to  
that the SCS treats the \(O_{c}\) of the CCS as \(O^{\prime}_{c}\), and the beam, which points from \(O^{\prime}_{c}\) to the center of \(C\), matches the \(z^{\prime}_{c}\) axis. The \(x^{\prime}_{c}\) and \(y^{\prime}_{c}\) axes conform to the right-handed coordinate system. Here, \(\mathbf{R}\) is used to denote the rotation matrix from the SCS to the CCS as well as the conversion matrix:  
\[\mathbf{P}_{Qc}=\mathbf{R}\mathbf{P}^{\prime}_{Qc} \tag{11}\]  
By substituting Equation (11) into Equation (10), the following expression is obtained:  
\[\mathbf{P}^{\prime}_{Qc}\mathbf{R}^{-1}\mathbf{W}\mathbf{R}\mathbf{P}^{ \prime\text{T}}_{Qc}=0. \tag{12}\]  
\(\mathbf{R}^{-1}\mathbf{W}\mathbf{R}\) is established through diagonalization--i.e., \(\mathbf{R}^{-1}\mathbf{W}\mathbf{R}=\text{Diag}(\lambda_{1},\,\lambda_{2},\, \lambda_{3})\). In other words, after the eigenvalue decomposition of \(\mathbf{W},\,\mathbf{R}\) and the corresponding eigenvalues can be determined. Hence, the elliptic cone equation of \(Q_{i}\) in the SCS is written as:  
\[\lambda_{1}x^{\prime 2}_{Q}+\lambda_{2}y^{\prime 2}_{Q}+\lambda_{3}z^{\prime 2 }_{Q}=0 \tag{13}\]  
To satisfy these requirements, two operations must be performed. (1) The column vectors of \(\mathbf{R}\) should be unit-orthogonalized. (2) The order of the column vectors of \(\mathbf{R}\) must be adjusted based on the rules of \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\): (\(\lambda_{1}<\lambda_{2}<0\), \(\lambda_{3}>0\)) or (\(\lambda_{1}>\lambda_{2}>0\), \(\lambda_{3}<0\)). 3. Determining the parameters of two possible circles  
As mentioned above, the elliptic cone \(Q\) may be uniquely identified, and its equation is expressed by Equation (13) in the SCS. Hence, the determination of the circle position and orientation is equivalent to locating a plane that intersects the elliptic cone \(Q\) to form a circular ring with radius \(R\). In other words, the center coordinates of the circle ring are the positional parameter of circle \(C\) and the normal vector of the plane is the orientational parameter of circle \(C\). Nonetheless, according to the geometrical relation, there are two planes that satisfy this condition, as shown by the red and blue dotted circles in Figure 3. In Ref. [28], the following expressions of the center and normal vector of the two circle rings formed by the two planes are reported:  
\[\left\{ \begin{array}{l}(x^{\prime}_{o},y^{\prime}_{o},z^{\prime}_{o})=\left( \pm R\sqrt{\frac{\lambda_{3}|\lambda_{1}|}{|\lambda_{1}|}(|\lambda_{1}|+|\lambda _{3}|)},0,R\sqrt{\frac{|\lambda_{1}|}{|\lambda_{3}|}(|\lambda_{2}|+|\lambda_{ 3}|)}\right)\\ \left(n^{\prime}_{ox},n^{\prime}_{oy},n^{\prime}_{oz}\right)=\left(\pm\sqrt{ \frac{|\lambda_{1}|}{|\lambda_{1}|}-\frac{|\lambda_{2}|}{|\lambda_{1}|}},0,- \sqrt{\frac{|\lambda_{2}|+|\lambda_{3}|}{|\lambda_{1}|+|\lambda_{3}|}}\right) \end{array} \right. \tag{14}\]  
where \((x^{\prime}_{o},y^{\prime}_{o},z^{\prime}_{o})\) and \((n^{\prime}_{ox},n^{\prime}_{oy},n^{\prime}_{oz})\) represent the center and normal vector of the two circle rings in the SCS, respectively. To unify the coordinate system with the light plane, the latter is transformed to the CCS according to the following relationships:  
\[\left\{