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We introduce a new class of curves and surfaces by exploring multiple variations of non-uniform rational B-splines. These variations which are referred to as generalized non-uniform rational B-splines (GNURBS) serve as an alternative interactive shape design tool, and provide improved approximation abilities in certain applications. GNURBS are obtained by decoupling the weights associated with control points along different physical coordinates. This unexplored idea brings the possibility of treating the weights as additional degrees of freedoms. It will be seen that this proposed concept effectively improves the capability of NURBS, and circumvents its deficiencies in special applications. Further, it is proven that these new representations are merely disguised forms of classic NURBS, guaranteeing a strong theoretical foundation, and facilitating their utilization. A few numerical examples are presented which demonstrate superior approximation results of GNURBS compared to NURBS in both cases of smooth and non-smooth fields. Finally, in order to better demonstrate the behavior and abilities of GNURBS in comparison to NURBS, an interactive MATLAB toolbox has been developed and introduced.