We extend the concept of generalized NURBS (GNURBS), recently introduced by the authors for parametric curves, to bivariate parametric surfaces. These generalizations are obtained via either explicit or implicit decoupling of the weights along different physical coordinates. This decoupling allows for treating the weights as additional degrees of freedom in a wider range of applications compared to classic NURBS surfaces, providing additional flexibility and increased control. This proposed concept effectively improves the capability of NURBS and alleviates its deficiencies in certain applications. In particular, we will demonstrate that GNURBS can be effectively used for improved approximation of certain class of surfaces such as helicoids, revolved surfaces and minimal surfaces. It will also be established that these proposed generalizations can be exactly transformed to equivalent, but higher order, classic NURBS surfaces, ensuring a strong theoretical foundation. Finally, a comprehensive MATLAB toolbox, GNURBS3D-Lab, has been developed and introduced in order to better demonstrate the behavior and properties of GNURBS surfaces compared to classic NURBS.

我们将作者最近为参数曲线引入的广义 NURBS (GNURBS) 的概念扩展到二元参数曲面。这些概括是通过沿不同物理坐标的权重的显式或隐式解耦获得的。与经典 NURBS 曲面相比，这种解耦允许在更广泛的应用中将权重视为额外的自由度，从而提供额外的灵活性和增强的控制。这个提出的概念有效地提高了 NURBS 的能力并减轻了它在某些应用中的不足。特别是，我们将证明 GNURBS 可以有效地用于改进某些类曲面的近似，例如螺旋面、旋转曲面和最小曲面。还将确定这些提议的概括可以准确地转换为等效但更高阶的经典 NURBS 曲面，从而确保强大的理论基础。最后，开发并引入了一个全面的 MATLAB 工具箱 GNURBS3D-Lab，以便与经典 NURBS 相比，更好地展示 GNURBS 曲面的行为和属性。