Smooth transition and Gibbs oscillation minimization in a 7-point subdivision scheme with shape-control parameters for high smoothness

Computer graphics is a dynamic field that heavily relies on mathematical techniques. For instance, subdivision scheme is used to create smooth and visually appealing curves and surfaces of arbitrary topology. The primary focus of this study is to transform two 5-point binary subdivision schemes into a single 7-point binary subdivision scheme with shape control. We have merged two binary approximating schemes that were constructed using the uniform B-spline basis function and the Lagrange basis function into a new subdivision scheme. It has been demonstrated that, for fixed values of the global shape control parameters, the curves generated by the proposed 7-point binary subdivision scheme maintain $C^6$ continuity everywhere. Furthermore, a brief discussion on the analysis of the Gibbs phenomenon in the new subdivision scheme has been presented. This is also a reminder of the challenges and intricacies involved in computer graphics and geometric modeling.