The New Curve and Surface Construction Approach Based on Generalized Bernstein Basis Functions With a Shape Parameter

Abstract

Extended Chebyshev spaces (ECT-spaces) are of significant importance and interest from a design perspective, as they provide valuable shape parameters that enhance control. In this investigation, we introduce a recursively defined generalized Bernstein basis (GB-basis) based on the blossom approach, together with a closed-form expression. This basis serves as a natural extension of the classical Bernstein basis functions of degree $n$. The new GB-basis extends the Bernstein basis to a broader context involving non-polynomial bases, including fractional, exponential, and trigonometric functions. We prove that the new GB-basis possesses shape-preserving properties, which play a fundamental role in computer-aided geometric design (CAGD) and computer graphics. A new corner-cutting algorithm is developed to efficiently compute the corresponding generalized Bézier curves (GB-curves), rational curves (RGB-curves), and surfaces (GB-surfaces). Moreover, the shape parameter can predictably adjust the shape of the new curves and surfaces without altering the control points. As the shape parameter increases, the resulting curves and surfaces approach the control polygon or control net in a predictable manner. To address challenges in creating complex curves, we derive necessary and sufficient conditions for $C^1$ continuity. Additionally, the GB-basis enables the construction of sharp-pointed curves. Experimental results demonstrate the superiority and efficiency of the new GB-basis over the classical Bernstein basis.