## 2 3D Burden Surface Feature Parameter Definition
### 3D Burden Surface Feature Parameter Definition  
[PERSON] proposed three 3D burden surface definition methods: linear definition, ring definition, and sector definition. However, this definition method still follows the defined strategy of the burden line, which has certain limitations. The 3D burden surface is an irregular spatial surface. At present, no researchers have systematically proposed a clear method for defining this. Based on the phased array radar point cloud data, a 3D burden surface feature definition method is proposed. The extracted feature parameters can reflect the shape of the burden surface well, which is helpful for the operator to understand the internal charge operation of the BF. Combined with the experience of on-site experts, the definition of burden surface features should be universal. Eighteen surface feature parameters were determined, as shown in Figure 2. In the figure, the spatial 3D burden surface is divided into three areas: the central coke area \(\Psi_{c}\), the funnel area \(\Psi_{f}\), and the platform area \(\Psi_{p}\). The spatial 3D burden surface shape is fitted by the three-segment ruled surface method. Once the equations of the highest point \(\left(c_{x},c_{y},c_{z}\right)\) and the three characteristic curves \(\Gamma_{d}\), \(\Gamma_{u}\), and \(\Gamma_{o}\) are determined, the spatial burden surface equation can be determined [18]. Figure 1: A high-resolution image of the burden surface. Using the parametric equation of the space curve to correlate the curve equations of two dimensions, the equations of the characteristic curves \(\Gamma_{d}\), \(\Gamma_{u}\), and \(\Gamma_{o}\) are obtained:  
\[\Gamma_{d}:\left\{ \begin{array}{l}\mathrm{X}_{d}(u)=x_{d}+R_{d}\cos u\\ \mathrm{Y}_{d}(u)=y_{d}+R_{d}\sin u\\ Z_{d}(u)=a_{1}u^{3}+a_{2}u^{2}+a_{3}u+a_{4}\\ \end{array} \right.\] \[\Gamma_{u}:\left\{ \begin{array}{l}\mathrm{X}_{u}(u)=x_{u}+R_{u} \cos u\\ \mathrm{Y}_{u}(u)=y_{u}+R_{u}\sin u\\ Z_{u}(u)=a_{5}u^{3}+a_{6}u^{2}+a_{7}u+a_{8}\\ \end{array} \right. \tag{1}\] \[\Gamma_{o}:\left\{ \begin{array}{l}\mathrm{X}_{o}(u)=x_{o}+R_{o} \cos u\\ \mathrm{Y}_{o}(u)=y_{o}+R_{o}\sin u\\ Z_{o}(u)=a_{9}u^{3}+a_{10}u^{2}+a_{11}u+a_{12}\\ \end{array} \right.\]  
The parametric equations of three region surfaces \(\Psi_{c}\), \(\Psi_{f}\), and \(\Psi_{p}\) are as follows:  
\[\Psi_{c}:\left\{ \begin{array}{l}\mathrm{X}_{c}(u,v)=(1-v)c_{x}+vX_{d}(u)\\ Y_{c}(u,v)=(1-v)c_{y}+vY_{d}(u)\\ Z_{c}(u,v)=(1-v)c_{z}+vZ_{d}(u)\\ \end{array} \right.\] \[\Psi_{f}:\left\{ \begin{array}{l}\mathrm{X}_{f}(u,v)=(1-v)X_{d}(u)+ vX_{u}(u)\\ Y_{f}(u,v)=(1-v)Y_{d}(u)+vY_{u}(u)\\ Z_{f}(u,v)=(1-v)Z_{d}(u)+vZ_{u}(u)\\ \end{array} \right. \tag{2}\] \[\Psi_{p}:\left\{ \begin{array}{l}\mathrm{X}_{p}(u,v)=(1-v)X_{u}(u)+ vX_{o}(u)\\ Y_{p}(u,v)=(1-v)Y_{u}(u)+vY_{o}(u)\\ Z_{p}(u,v)=(1-v)Z_{u}(u)+vZ_{o}(u)\\ \end{array} \right.\]  
The complete burden surface equation obtained by fitting is \(\Psi=\Psi_{c}+\Psi_{f}+\Psi_{p}\). The results of the area surface fitting are shown in Figure 3.  
Figure 2: 3D burden surface feature parameters. After the burden surface fitting process is completed, 18 feature parameters are extracted. The highest point \(C\big{(}c_{x},c_{y},c_{z}\big{)}\) in the central focal area is directly obtained from the point cloud data. The center points \(D(x_{d},y_{d})\), \(U(x_{u},y_{u})\), and \(O(x_{o},y_{o})\) and the radii \(R_{d}\), \(R_{u}\), and \(R_{o}\) are obtained by the least square method, and the calculation formulas of the remaining eight features are as follows.