Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.