Geometric modeling of some engineering GBT-Bézier surfaces with shape parameters and their applications

This study is based on some $C^{1}$ , $C^{2}$ , and $C^{3}$ continuous computer-based surfaces that are modeled by using generalized blended trigonometric Bezier (shortly, GBT-Bezier) curves with shape parameters. Initially, generalized blended trigonometric Bernstein-like (shortly, GBTB) basis functions with two shape parameters are derived in explicit expression which satisfied the basic geometric features of the traditional Bernstein basis functions. Moreover, the GBT-Bezier curves and tensor product GBT-Bezier surfaces with two shape parameters are also presented. All geometric features of the proposed GBT-Bezier curves and surfaces are similar to the traditional Bezier curves and surfaces, but the shape-adjustment is the additional feature that the traditional Bezier curves and surfaces do not hold. Finally, a class of some complex computer-based engineering surfaces via GBT-Bezier curves with shape parameters is presented. In addition, two adjacent GBT-Bezier surfaces segments are connected by higher $C^{2}$ and $C^{3}$ continuity constraints than the existing only $C^{1}$ shape adjustable Bezier surfaces. Some practical examples are provided to show the efficiency of the proposed scheme and to prove it as another powerful way for the construction and modeling of various complex composite computer-based engineering surfaces using higher-order continuities.