# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 2 SQ-Coons Surface
### Geometric Properties  
From this construction process of SQ-Coons surface \(S\), it could be concluded that except for the 12 control points of the boundary curves, the parameters of \(S\) include only four shape parameters--\(\alpha_{u},\gamma_{u}\), \(\alpha_{v}\) and \(\gamma_{v}\). Therefore, its boundaries are completely interpolated in the four traditional Bezier curves, \(p_{u0}\), \(p_{u1},p_{v0}\), \(p_{v1}\), and are unaffected by the shape parameters, while the shape of the surface's central part is determined only by shape parameters. For the surface displayed in Figure 3, in its \(u\)-direction, the surface approximates the virtual control points (as explained above, these points do not really exist in the definition of SQ-Coons surface) \(p_{11}\) and \(p_{12}\) gradually with the increase of \(\alpha_{u}\), and moves away from \(p_{11}\) and \(p_{12}\) with the decrease of \(\alpha_{u}\); similarly, it also approximates the virtual control points \(p_{21}\) and \(p_{22}\) with the increase of \(\gamma_{u}\), and moves away from \(p_{21}\) and \(p_{22}\) with the decrease of \(\gamma_{u}\). Analogously, in its \(v\)-direction, the values of \(\alpha_{v}\) and \(\gamma_{v}\) affect surface approximation or distance from the virtual control points \(p_{11}\), \(p_{21}\) and \(p_{12}\), \(p_{22}\). Figure 3: The modelling of the surface central part could be affected by the shape parameters, so as to be close to or far from the virtual control points. Figure 2: In order to generate the shape-adjustable surface proposed in this paper, three special surfaces are constructed: (**a**) The surface \(R_{1}\) of Bézier basis functions in its \(u\)-direction and CE-Bézier basis functions in its \(v\)-direction; (**b**) The surface \(R_{2}\) of CE-Bézier basis functions in its \(u\)-direction and Bézier basis functions in its \(v\)-direction; (**c**) The Coons surface \(S\) constructed by four CE-Bézier curves. Therefore, in the actual modeling design process, after determining the boundaries of the surface, the surface details could be adjusted in depth and meticulously through the shape parameters. The shape change of the surface can be described by the change of mean curvature (the average value of maximum curvature and minimum curvature at a point on a surface). In this paper, the mean curvature nephogram is adopted to show the slight shape differences of a surface. For the surface with the same four boundary curves in Figure 3, when \(\alpha_{v}\) and \(\gamma_{v}\) are both valued as 1, \(\alpha_{u}\) and \(\gamma_{u}\) are modified in the range of \(1\pm 4\); the shape change of the surface is illustrated by the mean curvature nephogram in Figure 4. As shown in the legend, the mean curvature varies from 0.1 to 0.7. It could be seen that all these curvature changes follow the surface modelling trend determined by boundary curves. Similarly, the change of parameters \(\alpha_{v}\) and \(\gamma_{v}\) would also adjust the surface shape, but in another direction. Moreover, the surface would degenerate into a common Coons surface when the four shape parameters are all valued as 1; when two shape parameters in the same direction are set as 1, the surface would transform into the surface expounded in Figure 2a or Figure 2b. Compared with other types of parametric surfaces including the Bezier surface, NURBS surface, CE-Bezier surface (or other parametric shape-adjustable Bezier surfaces), bilinearly blended Coons surface, bicubically blended Coons surface and extended Coons surface with shape parameters, the differences in properties and characteristics of the SQ-Coons surface are shown in Table 1. Among them, the SQ-Coons surface requires the least control information, which includes the least number of control points and has no need for cross-boundary derivatives. In addition, the surface could be interpolated at the boundaries and only the central part could be controlled by shape parameters.