## 3 Description of the Method
### Definition of the Stations of the Planing Surface
#### 3.2.2. Equations and Solution of the Stations of the Planing Surface  
Along the longitudinal range \([0,Lc]\) of the keel curve, equidistant coordinate values are selected. Within each segment, the cubic Bezier curves are utilized to construct the sliding surface station curves as shown by the orange curve in Figure 10.  
\[\mathbf{S}_{i}(u)=B_{0,3}(u)\cdot\mathbf{Q}_{ki}+B_{1,3}(u)\cdot\mathbf{C}_{1,i} +B_{23}(u)\cdot\mathbf{C}_{2,i}+B_{3,3}(u)\cdot\mathbf{Q}_{ci} \tag{20}\]  
For the analysis of the \(i\)-th station, by substituting \(x_{i}\) into the segmented keel curve in Equation (20), the intersection point \(\mathbf{Q}_{ki}\) of the keel curve and the cross-section can be determined. By substituting \(x_{i}\) into the horizontal projection of the inner chine curve in Equation (15), the corresponding abscissa \(y_{Qi}\) of \(\mathbf{Q}_{ci}\) is evaluated. Hence, the chord length can be evaluated by the following: \(l_{i}=y_{Qi}/\cos(\theta_{i})\). For ease of analysis, a local coordinate system \(\mathbf{Q}_{ki}\)-\(x_{i}y_{i}z_{i}\) is established, with \(\mathbf{Q}_{ki}\) as the origin. In this local coordinate system, the \(x\)-axis aligns with the hull coordinate system, and the \(y\)-axis aligns with the chord line. Let the rotation matrix around the \(x\)-axis be denoted as \(\mathbf{R}_{gi}\). The relationship between the coordinates of an arbitrary point \(\mathbf{P}\) in the hull coordinate system and its coordinates \(\mathbf{P}^{\prime}\) in the local coordinate system \(\mathbf{Q}_{ki}\)-\(x_{i}y_{i}z_{i}\) is as follows:  
\[\left\{\begin{array}{l}\mathbf{P}=\mathbf{P}^{\prime}\cdot\mathbf{R}_{gi}+ \mathbf{Q}_{ki}\\ \mathbf{P}^{\prime}=(\mathbf{P}-\mathbf{Q}_{ki})\cdot\mathbf{R}_{gi}^{-1} \end{array}\right. \tag{21}\]  
The \(i\)-th station curve is analyzed in the local coordinate system \(\mathbf{Q}_{ki}\)-\(x_{i}y_{i}z_{i}\). In this case, with \(\mathbf{Q}_{ki}\) as the origin, the coordinates of \(\mathbf{Q}_{ci}\) are \((0J_{i}0)\). The control points \(\mathbf{C}_{1,i}\) and \(\mathbf{C}_{2,i}\) are sought to satisfy the following parameter conditions: at the extremum point \(\mathbf{P}_{mi}\), the first derivative of the curve's \(Z\) component, \(Z^{\prime}(\mathbf{u})\), is equal to zero, and the z-coordinate at the extremum point \(\mathbf{P}_{mi}\) is denoted as \(z_{pmi}\). Additionally, the angle between the tangent vectors  
Figure 10: The station curve of the planing surface.  
at both ends and the chord line is denoted by \(\gamma_{ci}\) and \(\gamma_{ki}\). The system of equations can be derived as follows:  
\[\begin{bmatrix}\tan(\gamma_{ki})&-1&0&0\\ 0&0&\tan(\gamma_{ci})&1\\ 0&B_{1,3}^{\prime}(u_{pmi})&0&B_{2,3}^{\prime}(u_{pmi})\\ 0&B_{1,3}(u_{pmi})&0&B_{2,3}^{\prime}(u_{pmi})\\ \end{bmatrix}\begin{bmatrix}y_{\text{C1},i}\\ z_{\text{C1},i}\\ y_{\text{C2},i}\\ z_{\text{C2},i}\\ \end{bmatrix}=\begin{bmatrix}\tan(\gamma_{ci})\cdot l_{i}\\ 0\\ z_{\text{Pmi}}\\ \end{bmatrix} \tag{22}\]  
where the shape coefficients \(k_{s,i}\) and \(k_{m,i}\) are introduced to define \(u_{pmi}\) and \(z_{pmi}\) since \(u_{pmi}\) cannot be obtained via the chord-length parameterization method.  
\[\begin{cases}u_{pmi}=k_{s,i}+0.5\\ z_{pmi}=\frac{1}{2}\cdot(1+k_{m,i})\cdot\frac{\tan(\gamma_{ki})|\tan(\gamma_{ ci})|}{|\tan(\gamma_{ci})|+|\tan(\gamma_{ki})|}L_{i}\end{cases} \tag{23}\]  
The transverse curvature of the planing surface is determined by the shape coefficients (\(k_{s,i}\) and \(k_{m,j}\)) and the angles (\(\gamma_{ci}\) and \(\gamma_{ki}\)). By solving Equation (22), the local coordinate values of the control points are evaluated. By introducing these values to Equation (21), the global coordinate values of \(\mathbf{C}_{1,i}\) and \(\mathbf{C}_{2,i}\) are obtained. It can be proven that when the curved shape is either concave or convex, the value of \(k_{s,i}\) and \(k_{m,i}\) can be set equal to 0, resulting in an equivalent quadratic B-spline curve.