## 3 Description of the Method
### Definition of the Stations of the Planing Surface
#### 3.2.1 Definition of the Inner Chine Curve in Plan View  
The inner chine curve in plan view is illustrated in Figure 9, extending from \(\textbf{P}_{c0}\) at the stem to \(\textbf{P}_{c1}\) at the transmon through the widest point \(\textbf{P}_{cm}\). A cubic Bezier curve is utilized to represent the chine curve, where \(\textbf{C}_{c2}\) and \(\textbf{C}_{c3}\) represent the control points to be determined.  
\[\textbf{S}(u)=B_{0,3}(u)\cdot\textbf{P}_{c0}+B_{1,3}(u)\cdot\textbf{C}_{c1}+B_ {23}(u)\cdot\textbf{C}_{c2}+B_{3,3}(u)\cdot\textbf{P}_{c1} \tag{15}\]  
The main parameters used for the design geometry are as follows: stern width (\(B_{t}\)), maximum width of inner chine curve (\(B_{cm}\)), longitudinal length of inner chine curve (\(L_{c}\)), end tangent angle (\(\beta_{c}\)), horizontal projection area of the planing surface (\(A\)), and longitudinal coordinate of the centroid of horizontal projection (\(x_{Ac}\)). Since the area and centroid require some integrals to be calculated, it is not convenient to construct linear equations to get control points. By introducing the x-coordinate values \(x_{cm}\) and corresponding curve parameter \(u_{cm}\) values, as well as combining the constraints of terminal tangent angle and maximum width of chine, a system of equations for curve control points can be constructed, as follows:  
\[\begin{bmatrix}B_{1,3}(u_{cm})&0&B_{2,3}&0\\ 0&B_{1,3}(u_{cm})&0&B_{2,3}(u_{cm})\\ 0&-B_{1,3}^{\prime}(u_{cm})&0&-B_{2,3}^{\prime}(u_{cm})\\ 0&0&\tan(\beta_{c})&-1\\ \end{bmatrix}\begin{bmatrix}x_{C1}\\ y_{C1}\\ x_{C2}\\ y_{C2}\end{bmatrix}=\begin{bmatrix}x_{cm}-B_{3,3}(u_{cm})\cdot L_{c}\\ B_{cm}-B_{0,3}(u_{cm})\cdot B_{c0}\\ b_{0,3}^{\prime}(u_{cm})\cdot B_{c0}\\ \tan(\beta_{c})\cdot L_{c}\\ \end{bmatrix} \tag{16}\]  
By solving Equation (16), the control points \(\textbf{C}_{c2}\) and \(\textbf{C}_{c3}\) can be obtained. By substituting \(\textbf{C}_{c2}\) and \(\textbf{C}_{c3}\) into Equation (15), curve equation \(\textbf{S}(u)\) could be appropriately calculated, where \(X\big{(}u,x_{pcm},u_{cm}\big{)}\) and \(Y\big{(}u,x_{pcm},u_{cm}\big{)}\) represent the \(x\) and \(y\) components of \(\textbf{S}(u)\). Therefore, both the area \(A^{\prime}\) and centroid x-coordinates \(x^{\prime}_{AC}\) can be stated by \(x_{cm}\) and \(u_{cm}\), as follows:  
\[A^{\prime}(x_{cm},u_{cm})=2\int_{0}^{1}Y(u,x_{cm},u_{cm})X^{\prime}(u,x_{cm}, u_{cm})du \tag{17}\]  
\[x^{\prime}_{AC}(x_{cm},u_{cm})=\frac{2\int_{0}^{1}\!\!X(u,x_{cm},u_{cm})\cdot Y (u,x_{cm},u_{cm})\cdot X^{\prime}(u,x_{cm},u_{cm})du}{A(x_{cm},u_{cm})} \tag{18}\]  
Considering the design parameters of \(A\) and \(x_{Ac}\), the objective function is constructed as follows:  
\[\begin{cases}\quad target1=A(x_{pcm},u_{cm})-A\\ target2=x_{Ac}(x_{pcm},u_{cm})-x_{Ac}\end{cases} \tag{19}\]  
The non-dominated sorting genetic algorithm II (NSGA-II) [26] is a widely used multi-objective optimization algorithm that can effectively handle multiple conflicting objectives simultaneously. The NSGA-II is implemented to search for \(x_{cm}\) and \(u_{cm}\), aiming to minimize the objective function in Equation (19) and satisfy the area and centroid conditions. By substituting the obtained solutions into Equation (16), the numerical solution of the curve control points \(\textbf{C}_{c2}\) and \(\textbf{C}_{c3}\) can be obtained by satisfying the area and centroid requirements.  
Figure 9: The inner chine curve in plan view.