# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 3 Automobile Body Design with the SQ-Coons Surface
### Range of Shape Parameters  
The default value of the four shape parameters is 1, and at this point the SQ-Coons surface could degenerate into a traditional bilinearly blended Coons surface. Beyond that, shape parameters do not have a clear range of values. However, one SQ-Coons surface has four shape parameters,  
Figure 6: Effect of altering the control point on vehicle modeling: (**a**) Moving the control points of the windshield forward; (**b**) Moving the control points back. Figure 7: Effect of altering the shape parameters on hood surface: (**a**) \(\alpha_{u}\), \(\gamma_{u}\), \(\alpha_{v}\), \(\gamma_{v}=1\); (**b**) \(\alpha_{u}\), \(\gamma_{u}\), \(\alpha_{v}\), \(\gamma_{v}=-3\). and excessive value possibilities would greatly increase the number of sample groups in simulation or testing. Therefore, it is necessary to limit the range of parameters. For this paper, focusing on surface type in car form design, a parameter range limitation algorithm based on modeling design needs to be established. The requirement of this restriction is that the whole surface modeling should not change excessively when the shape parameters are altered. As an important indicator of surface shape description, curvature and its variation range could be used as the main factors to limit the change of shape parameters. The common curvatures of surfaces are Gauss curvature and mean curvature, which are determined by the maximum curvature and the minimum curvature at one point on a surface. However, as this research pays more attention to the curvature change in two directions (\(u\) and \(v\)) of one surface, the curvature of a surface could be transformed into the curvature of some key isocurves on surfaces. Based on this, a difference degree \(d\) of some key isocurves addressed in this paper can be proposed and regarded as the most important indicator to detect whether the surface is over-deformed. As indicated in Figure 8, three isocurves in both directions (\(u\) and \(v\)) (the values are \(u=0\), \(0.5\), \(1\) and \(v=0\), \(0.5\), \(1\)) are extracted. The curve \(k\) is calculated according to the following:  
\[k=\left[\left[\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}}\right]^{2}+\left[\frac{ \mathrm{d}^{2}y}{\mathrm{d}t^{2}}\right]^{2}+\left[\frac{\mathrm{d}^{2}z}{ \mathrm{d}t^{2}}\right]^{2}\right]^{1/2} \tag{11}\]  
From the characteristics of this SQ-Coons surface, the surface is completely interpolated on four boundary curves, which means that the curvatures of boundary curves, \(u=0\), \(1\) and \(v=0\), \(1\), would not change with the adjustment of shape parameters, while the curvatures of the other two isocurves, \(u=0.5\) and \(v=0.5\), would change because of the modification of shape parameters. Therefore, the difference degrees of curvatures on two isocurves and their corresponding boundary lines can be calculated and counted. Four difference degrees--\(d_{u0}\), \(d_{u1}\), \(d_{v0}\), and \(d_{v1}\)--in two directions--\(u\) and \(v\)) are defined as follows:  
\[\left\{ \begin{array}{l}d_{u0}=\frac{k_{u-0.5}-k_{u-0}}{k_{u-0}}\\ d_{u1}=\frac{k_{u-0.5}-k_{u-1}}{k_{u-1}}\\ d_{v0}=\frac{k_{u-0.5}-k_{v-0}}{k_{u-0}}\\ d_{v1}=\frac{k_{v-0.5}-k_{v-1}}{k_{v-1}}\end{array} \right. \tag{12}\]  
In order to determine the value of these four differences in common automobile models, the models of 50 saloon sedans--defined as a sample library and marked as M1-M50--were built through the model template proposed in this paper. Among them, M1-M15 are shown in Figure 9. The original model template is named M0. Figure 8: Some key isocurves of a surface are used to define the surface curvature: (**a**) The isocurves \(u=0\), \(0.5\), \(1\) in direction-\(u\); (**b**) The isocurves \(v=0\), \(0.5\), \(1\) in direction-\(v\).