Bezier Curves and Surfaces with the Generalized α-Bernstein Operator

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In the field of Computer-Aided Geometric Design (CAGD), a proper model can be achieved depending on certain characteristics of the predefined blending basis functions. The presence of these characteristics ensures the geometric properties necessary for a decent design. The objective of this study, therefore, is to examine the generalized α-Bernstein operator in the context of its potential classification as a novel blending type basis for the construction of Bézier-like curves and surfaces. First, a recursive definition of this basis is provided, along with its unique representation in terms of that for the classical Bernstein operator. Next, following these representations, the characteristics of the basis are discussed, and one shape parameter for α-Bezier curves is defined. In addition, by utilizing the recursive definition of the basis, a de Casteljau-like algorithm is provided such that a subdivision schema can be applied to the construction of the new α-Bezier curves. The parametric continuity constraints for C0, C1, and C2 are also established to join two α-Bezier curves. Finally, a set of cross-sectional engineering surfaces is designed to indicate that the generalized α-Bernstein operator, as a basis, is efficient and easy to implement for forming shape-adjustable designs.