Abstract The construction of the generalized Bezier model with shape parameters is one of the research hotspots in geometric modeling and CAGD. In this paper, a novel shape-adjustable generalized Bezier (or SG-Bezier, for short) surface of order (m, n) is introduced for the purpose to construct local and global shape controllable free-form complex surfaces. Meanwhile, some properties of SG-Bezier surfaces and the influence rules of shape parameters, as well as the constructions of special triangular and biangular SG-Bezier surfaces, are investigated. Furthermore, based on the terminal properties and linear independence of SG-Bernstein basis functions, the conditions for G1 and G2 continuity between two adjacent SG-Bezier surfaces are derived, and then simplified them by choosing appropriate shape parameters. Finally, the specific steps and applications of the smooth continuity for SG-Bezier surfaces are discussed. Modeling examples show that our methods in this paper are not only effective and can be performed easily, but also provide an alternative strategy for the construction of complex surfaces in engineering design.

中文翻译：

---

可调整形状的广义贝塞尔曲面：构造及其几何连续性条件

摘要 带形状参数的广义Bezier模型的构建是几何建模和CAGD的研究热点之一。在本文中，为了构造局部和全局形状可控的自由形式复杂曲面，引入了一种新的形状可调广义 Bezier（或简称 SG-Bezier）阶 (m, n) 曲面。同时，研究了SG-Bezier曲面的一些性质和形状参数的影响规律，以及特殊三角形和双角SG-Bezier曲面的构造。此外，基于SG-Bernstein基函数的终端性质和线性独立性，推导出相邻两个SG-Bezier曲面之间G1和G2连续性的条件，然后通过选择合适的形状参数对其进行简化。最后，讨论了SG-Bezier曲面平滑连续性的具体步骤和应用。建模实例表明，我们在本文中的方法不仅有效且易于执行，而且还为工程设计中复杂曲面的构建提供了一种替代策略。