Optimal control, construction, and analysis of λ-Bernstein Bézier surfaces for quasi-harmonic functional

In this article, we investigate a novel construction scheme for ?-Bernstein B?zier surfaces and illustrate it with biquadratic and bicubic cases to demonstrate their geometric characteristics and their applications. The primary objective is to investigate how the shape parameter ? improves control over surface smoothness and facilitates an optimal solution in surface design. We examine the geometric properties of these surfaces, including mean and Gaussian curvature, shape operator coefficients, and Gauss-Weingarten coefficients. We also analyze the extremal conditions for ?-Bernstein B?zier surfaces derived from the vanishing condition for the gradient of the quasi-harmonic functional. Integral formulations based on Bernstein polynomials enable precise computation of the vanishing gradient condition, allowing us to determine the constraints on interior control points in terms of known boundary control points. Graphical illustrations validate the approach by providing a better understanding of the geometric properties of these surfaces, including improved surface smoothness and design flexibility. They effectively showcase the behavior of ?-Bernstein polynomials and their corresponding surfaces. Computational results demonstrate the effectiveness of this method for applications in computer graphics, computational geometry, computer science, and engineering, offering a robust framework for analyzing and generating optimal surfaces, contributing to advancements across various scientific disciplines.