# A Novel Shape-Adjustable Surface and Its Applications in Car Design
## 1 Introduction
The basis geometry theory of CAD technology is computer aided geometric design (CAGD), of which the classic tools and basic elements are parametric curves and surfaces. Bezier, B-spline and NUBRS, the major components among them, were proposed decades ago and are widely used in all kinds of general-purpose CAD software. However, in recent years, these traditional curves or surfaces struggle to meet the practical requirements in the design or engineering field [13]. This is because the shape cannot be adjusted by additional parameters except the control points, or they cannot accurately represent conic curves and surfaces, or the calculation is cumbersome. Therefore, in order to overcome these problems, researchers have proposed many novel parameterized Bezier-like curves and surfaces, and constantly expanded their application prospects. [PERSON] et al. [14] proposed a new transformation algorithm to define the C-Bezier curve with one shape parameter introduced as a separated form including a Bezier part and a trigonometric part, in order to represent some regular curves such as a cycloid or traditional Bezier curve by the C-Bezier curve. By introducing two shape parameters, [PERSON] et al. [15] constructed a cubic trigonometric Bezier (T-Bezier) curve based on a cubic Bezier curve, which has strong geometric properties and could exactly represent ellipses. [PERSON] et al. [13] presented a shape-adjustable generalized Bezier (SG-Bezier) curve and surface with global and local shape parameters by extending classical Bernstein basis functions. In their subsequent research [16], they studied in depth the multiple geometric properties of the SG-Bezier curve, which includes the conditions for continuity between two curves, the method to smooth two curves, and the influence of shape parameters on curve form. Adjusting Beta function, [PERSON] et al. [17] proposed a scheme for generating new types of curves and triangular patches, in which shape parameters are introduced on the basis of preserving the correlation characteristics of the Bezier curve and surface, and this improves the flexibility of its application. [PERSON] et al. [18] extended the CE-Bezier surface, a previously proposed bicubic surface with better flexibility, shape adjustability and approximation, to degree m*n, which is termed the GE-Bezier surface. Furthermore, to solve the problem of whether the shape parameterschange after the segmentation of the CE-Bezier curve, [PERSON] [19] conducted a thorough study and proposed two different segmentation algorithms. However, all the parameterized Bezier-like surfaces mentioned above have a common disadvantage in the actual product design process, that is, the boundaries of these surfaces are also deformed and affected by shape parameters and cannot be interpolated on their initial boundary curves. Therefore, the designer cannot adjust the surface shape while maintaining the boundary shape, and vice versa, which is against the goal of efficient design. In the CAGD field, another vital surface, Coons surface, has the advantage of overcoming this defect. Coons surface was proposed as early as the mid-1960s, and its characteristic is that the surface shape is defined by boundary curves and related parameters. There are two types of Coons surface. The first type is bilinearly blended Coons patch, in which the shape is completely interpolated into its four boundaries and cannot be adjusted. The construction of this surface is the superposition of three surfaces, in which the first and second are ruled surfaces in different directions, and the third is a bilinear tensor product surface. The second type is bicubically blended Coons, which is based on the first type and interpolates the cross-boundary derivatives in addition to the boundaries, by adding a set of blending functions. Similar to various Bezier-like surfaces, scholars introduced shape parameters to improve the modelling adjustability of the second type of Coons patch. Introducing trigonometric Hermite basis functions, [PERSON] et al.