# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 2 SQ-Coons Surface  
### Mathematical Definition  
This study implements a shape-adjustable quadrilateral surface interpolated with boundary curves, in which the basic function of the CE-Bezier surface and the construction of the Coons surface are combined. Therefore, it also has the preponderance of these two surfaces. To construct a bilinearly blended Coons surface, only the control points of its boundary curve need to be provided. However, when establishing other parametric surfaces (e.g., Bezier surface), the coordinates of the control points at the center of the surface are also needed. Therefore, in this paper, the control point information of the non-boundary part is obtained by converting the Coons surface into the bicubic Bezier surface. As illustrated in Figure 1a, four cubic Bezier curves, connected end to end, are constructed as (\(I,j=0,1,2,3\)):  
\[\left\{ \begin{array}{l}p_{u0}=\sum\limits_{i=0}^{3}B_{i,3}(u)\cdot p_{0i}\\ p_{u1}=\sum\limits_{i=0}^{3}B_{i,3}(u)\cdot p_{3i}\\ p_{v0}=\sum\limits_{j=0}^{3}B_{j,3}(v)\cdot p_{j0}\\ p_{v1}=\sum\limits_{j=0}^{3}B_{j,3}(v)\cdot p_{j3}\end{array} \right. \tag{1}\]  
where \(B_{i,3}(t)\) (\(i=0,1,2,3\)) is the traditional cubic Bezier basis functions:  
\[\left\{ \begin{array}{l}B_{0,3}(t)=\left(1-t\right)^{3}\\ B_{1,3}(t)=3t(1-t)^{2}\\ B_{2,3}(t)=3t^{2}(1-t)\\ B_{3,3}(t)=t^{3}\end{array} \right. \tag{2}\]  
The bilinearly blended Coons patch \(P(u,v)\) in Figure 1b surrounded by the four curves, \(p_{u0}\), \(p_{u1}\), \(p_{v0}\), \(p_{v1}\), is created as:  
\[P(u,v)=-\left[ \begin{array}{ccc}-1&u&1-u\end{array} \right]\left[ \begin{array} []{ccc}0&p_{u0}&p_{u1}\\ p_{0v}&p_{00}&p_{03}\\ p_{1v}&p_{30}&p_{33}\end{array} \right]\left[ \begin{array}{ccc}-1\\ v\\ 1-v\end{array} \right] \tag{3}\]  
After converting it into the bicubic Bezier surface, the 16 new control points \(p_{ij}\) (\(i,j=0,1,2,3\)) in Figure 1c are calculated as follows:  
\[p_{ij}=\left[ \begin{array}{ccc}p_{00}&p_{01}&p_{02}&p_{03}\\ p_{10}&p_{11}&p_{12}&p_{13}\\ p_{20}&p_{21}&p_{22}&p_{23}\\ p_{30}&p_{31}&p_{32}&p_{33}\end{array} \right]=\sum\limits_{i=0}^{3}\sum \limits_{j=0}^{3}\frac{P(u,v)}{B_{i,3}(u)\cdot B_{j,3}(u)} \tag{4}\]  
Besides the 12 control points on four boundary curves, the remaining four control points (\(p_{11}\), \(p_{12}\), \(p_{21}\), \(p_{22}\), which are termed virtual control points, as they do not belong to the input parameters of the  
Figure 1: The approaches to obtain four virtual control points besides the boundary control points by transforming the Coons surface into the Bézier surface: (**a**) Four boundary curves and 12 control points on the Bézier surface; (**b**) The Coons surface generated accroading to the boundary curves; (**c**) 16 control points of the Bézier surface converted from the Coons surface. new surface, but only to the intermediate process of generation) are located in the center of the surface. It could be concluded that that all control points of a simple modeling surface could be calculated from the boundaries. The basis functions of CE-Bezier are as follows:  
\[\left\{ \begin{array}{l}C_{0,3}(t)=(1-\alpha t)(1-t)^{2}\\ C_{1,3}(t)=(2+\alpha)(1-t)^{2}t\\ C_{2,3}(t)=(2+\gamma)t^{2}(1-t)\\ C_{3,3}(t)=(1-\gamma+\gamma t)t^{2}\end{array} \right. \tag{5}\]  
where \(\alpha\) and \(\gamma\) are the shape parameters, and the CE-Bezier basis functions could degenerate into cubic Bezier basis functions when \(\alpha\) and \(\gamma\) are valued as 1. In addition, the most significant feature of the CE-Bezier basis function is that the shape of the CE-Bezier curve or surface could be modified by adjusting the shape parameters, without changing the control points at all. Based on the control points \(p_{ij}\) (including 12 real control points and 4 virtual control points) and the basis functions of CE-Bezier, a special surface is established, one direction of which is according to the traditional Bezier basis functions, the other direction is based on the CE-Bezier basis functions.