## 3 Description of the Method
### Definition of the Keel Curve
#### 3.1.1 Definition of the Bow Curved Lift Section  
The curved lift section of the bow is mainly related to its navigation performance. The steeply sloping shape allows the bow to lift at low speeds, which is of great advantage for planing. However, from the perspective of spray performance requirements, a smaller angle between the keel curve and the waterline is desirable to improve spray performance. To balance these two requirements, the design of the bow of a planing boat usually possesses a relatively flat shape below the waterline and a steeper shape above the waterline. To represent this curve, a cubic Bezier curve is taken into account, with \(\mathbf{P}_{k1}\) and \(\mathbf{P}_{k3}\) as curve endpoints used. \(\mathbf{C}_{k1}(x_{ck1},0,z_{ck1})\) and \(\mathbf{C}_{k2}(x_{ck2},0,z_{ck2})\) were the control points to be solved.  
\[\mathbf{S}_{k2}(u)=B_{0,3}(u)\cdot\mathbf{P}_{k1}+B_{1,3}(u)\cdot\mathbf{C}_{k 1}+B_{23}(u)\cdot\mathbf{C}_{k2}+B_{3,3}(u)\cdot\mathbf{P}_{k3} \tag{10}\]  
As illustrated in Figure 6, the chord line \(\mathbf{P}_{k1}\mathbf{P}_{k3}\) is constructed, and the point \(\mathbf{P}_{k2}\) represents the furthest point of the curve from the chord \(\mathbf{P}_{k1}\mathbf{P}_{k3}\). The curve \(\mathbf{S}_{k2}(\text{u})\) satisfies the following conditions: (i) it passes through point \(\mathbf{P}_{k2}\); (ii) the slope of the curve at point \(\mathbf{P}_{k2}\) is equal to the slope of the chord line; and (iii) at point \(\mathbf{P}_{k1}\), it is tangent to the first segment. Therefore, a system of equations can be established and solved to obtain the coordinates of the control points of the curve, as follows:  
\[\begin{bmatrix}B_{1,3}\big{(}u_{p12}\big{)}&0&B_{2,3}\big{(}u_{p2}\big{)}&0\\ 0&B_{1,3}\big{(}u_{p2}\big{)}&0&B_{2,3}\big{(}u_{p2}\big{)}\\ \eta_{1}B_{1,3}^{\prime}&-B_{1,3}^{\prime}\big{(}u_{p2}\big{)}&\eta_{1}B_{2,3} ^{\prime}\big{(}u_{p2}\big{)}&-B_{2,3}^{\prime}\big{(}u_{p2}\big{)}&-B_{2,3}^{ \prime}\big{(}u_{p2}\big{)}\\ \eta_{2}&-1&0&0\end{bmatrix}\begin{bmatrix}x_{C11}\\ z_{C1}\\ z_{C12}\\ z_{C12}\end{bmatrix}=\begin{bmatrix}x_{P22}-B_{0,3}\big{(}u_{p12}\big{)}\cdot L _{0}-B_{3,3}\big{(}u_{p2}\big{)}\cdot L_{c}\\ z_{P22}-B_{3,3}\big{(}u_{p2}\big{)}\cdot H_{c}\\ B_{3,3}^{\prime}\big{(}u_{p2}\big{)}\cdot H_{c}-\eta_{1}B_{0,3}^{\prime} \big{(}u_{p2}\big{)}\cdot L_{0}-\eta_{1}B_{3,3}^{\prime}\big{(}u_{p2}\big{)} \cdot L_{c}\\ \eta_{2}\cdot L_{0}\end{bmatrix} \tag{11}\]  
where \(\eta_{1}\) represents the slope of the chord line \(\mathbf{P}_{k1}\mathbf{P}_{k3}\), \(\eta_{2}\) denotes the slope of the tangent vector at point \(\mathbf{P}_{k1}\), and \(u_{p2}\) denotes the corresponding curve parameter value at point \(\mathbf{P}_{k2}\). \(u_{pk2}\) can be calculated by the chord-length parameterization approach.  
To express the local geometric relationships in terms of relatively independent design parameters, point \(\mathbf{P}_{k2}\) should be transformed into a relative position form using parameters \(\lambda_{k1}\) and \(\lambda_{k2}\), as in the following form:  
\[\mathbf{P}_{k2}=\mathbf{P}_{k1}+(1+\lambda_{k1})\cdot\mathbf{n}_{k1}+(1+\lambda_ {k2})\cdot\mathbf{n}_{k2} \tag{12}\]  
Figure 5: The keel curve in lateral view.  
Figure 6: The bow curved lift section of keel curve.  
In addition, \(\mathbf{n}_{k1}=0.5\cdot\mathbf{P}_{k1}\mathbf{P}_{k3}\), and \(\mathbf{n}_{k2}=0.5\tan(\alpha_{k1})\cdot|\mathbf{n}_{k1}|\mathbf{n}_{\perp}\) where \(\mathbf{n}_{\perp}\) denotes the unit vector perpendicular to \(\mathbf{n}_{k1}\). When both parameters \(\lambda_{k1}\) and \(\lambda_{k2}\) are set equal to 0, it can be readily proven that the curve would be equivalent to a quadratic B-spline curve.