Using the Blossom method in the quasi extended Chebyshev space, we first construct a class of generalized cubic Bernstein functions in the generalized cubic polynomial space \(\{1,3t^2 - 2t^3, (1-t)^2 \lambda (t), t^2 \mu (t)\}\) . Our work includes many previous works as special cases by selecting specific shape functions \(\lambda (t) \) and \(\mu (t)\) . The resulting generalized cubic Bézier curves and their properties are discussed. The corner cutting method is also proposed to compute curves efficiently and stably. A generalized Bernstein operator is given and spectral analysis on a specific case is carried out. The results show that the obtained generalized Bézier curves can better fit the control polygon compared to the Bézier curves. We further deduce the closed-form solution of generalized cubic B-splines that possess local shape functions. Some important properties are presented, including local support property, nonnegativity, partition of unity, total positivity property, \(C^2\) continuity, linear independence and so on. Some numerical examples shows that the resulting generalized cubic B-spline curves can approximate the control polygon more flexibly.
