## 3 Smoothing Methodology  
### Cubic Bezier Curve  
A cubic Bezier curve is a continuous curvature curve with the smallest order and is given by  
\[\mathbf{a}(t)= (1-t)^{3}\mathbf{a}_{0}+3t(1-t)^{2}\mathbf{a}_{1} \tag{15}\] \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]  
where \((\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{a}_{3})\) are control points. This curve passes through points \(\mathbf{a}_{0}\) and \(\mathbf{a}_{3}\), and passes closely through the two middle points \(\mathbf{a}_{1}\) and \(\mathbf{a}_{2}\). Now, suppose that all four control points are on the same plane, and three of them are on the same straight line, i.e.,  
Figure 8: Two planes and intersection line. \[\mathbf{a}_{1} =\mathbf{a}_{0}+d_{1}\mathbf{e}_{in} \tag{16}\] \[\mathbf{a}_{2} =\mathbf{a}_{1}+d_{2}\mathbf{e}_{in}\] \[\mathbf{a}_{3} =\mathbf{a}_{2}+d_{3}\mathbf{e}_{mid}\]  
where \(\mathbf{e}_{in}\) and \(\mathbf{e}_{mid}\) are unit direction vectors, and \(d_{1},d_{2}\), and \(d_{3}\) denote the separation distances between control points. The curvature equation of the curve is given by  
\[\kappa(t)=\frac{\left\|\mathbf{a}^{\prime}(t)\times\mathbf{a}^{\prime\prime}( t)\right\|_{2}}{\left\|\mathbf{a}^{\prime}(t)\right\|_{2}^{3}}. \tag{17}\]  
The curvature is computed as \(\kappa(0)=0\) at \(\mathbf{a}_{0}=\mathbf{a}(0)\) and is monotonically increasing as \(t\) goes. For the curvature to be maximized at \(\mathbf{a}_{3}\) with its derivative zero, the separation distance must have the following values, as in [7]:  
\[d_{2} =0.346L,\ d_{1}=0.58d_{2} \tag{18}\] \[d_{3} =1.31d_{2}\cos\beta,\ \ 0<\beta<\frac{\pi}{2}\]  
where \(L\) is a design parameter that can be expressed as a function of the maximum allowed curvature, and \(\beta\) is the angle between \(\mathbf{e}_{in}\) and \(\mathbf{e}_{mid}\). The maximum curvature occurs at \(\mathbf{a}_{3}=\mathbf{a}(1)\) and is computed as  
\[\kappa_{max}=\kappa(1)=\frac{2d_{2}\sin\beta}{3d_{3}^{2}} \tag{19}\]  
With the Bezier curve described above, two curves can be concatenated to generate a continuous curvature path between two successive line segments in which the curvature is zero at both the start and end points located on each line segment. For two Bezier curves to be connected smoothly and to make maximum curvature occur at the connecting point, the last and first controls of both curves must coincide, and their derivatives of the curvature must be zero. The first Bezier curve and its control points are defined by (15) and (16), and the second curve is defined symmetrically, as shown in \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]0  
\[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]1 \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]2  
where \((\mathbf{b}_{0},\mathbf{b}_{1},\mathbf{b}_{2},\mathbf{b}_{3})\) are control points and defined by the same way to their corresponding counter parts \((\mathbf{a}_{0},\mathbf{a}_{1},\mathbf{a}_{2},\mathbf{a}_{3})\). \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]3 \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]4 \[+3t^{2}(1-t)\mathbf{a}_{2}+t^{3}\mathbf{a}_{3},\ \ 0\leq t\leq 1\]5  
In Figure 9, \(\mathbf{e}_{in}\) and \(\mathbf{e}_{out}\) are unit vectors representing the direction along each line segment, \(\mathbf{e}_{mid}\) is a unit vector along the line connecting \(\mathbf{a}_{2}\) and \(\mathbf{b}_{2}\), \(\mathbf{m}\) is the intersection point of two line segments, and \(\theta\) is the turning angle between \(\mathbf{e}_{in}\) and \(\mathbf{e}_{out}\), which is \(\theta=2\beta\) by the geometric relationship. It is easily shown that \(L\) is a distance from intersection point \(\mathbf{m}\) to \(\mathbf{b}_{0}\) or \(\mathbf{c}_{0}\). Points \(\mathbf{a}_{3}\) and \(\mathbf{b}_{3}\) are located in the center of the line connecting \(\mathbf{a}_{2}\) and \(\mathbf{b}_{2}\). The control points \(\mathbf{a}_{3}\) and \(\mathbf{b}_{3}\) are connecting points and must be equal.