Quantum (q, h)-Bézier surfaces based on bivariate (q, h)-blossoming

Abstract

We introduce the (q, h)-blossom of bivariate polynomials, and we define the bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces on rectangular domains using the tensor product. Using the (q, h)-blossom, we construct recursive evaluation algorithms for (q, h)-Bézier surfaces and we derive a dual functional property, a Marsden identity, and a number of other properties for bivariate (q, h)-Bernstein polynomials and (q, h)-Bézier surfaces. We develop a subdivision algorithm for (q, h)-Bézier surfaces with a geometric rate of convergence. Recursive evaluation algorithms for quantum (q, h)-partial derivatives of bivariate polynomials are also derived.

Degree Reduction of Q-Bézier Curves via Squirrel Search Algorithm

“…Hu et al [19] derived G 1 and G 2 continuity conditions of adjacent (m, n)-degree Q-Bézier surfaces with shape parameters. Jegdić et al [20] defined a (q, h)-blossom operator of bivariate Bernstein polynomials, and presented the corresponding rectangular tensor product (q, h)-Bézier surfaces. Recently, Delgado [21] investigated the geometric properties and algorithms for rational Q-Bézier curves and surfaces.…”

ALGORITHMS AND IDENTITIES FOR BIVARIATE $(h_1, h_2)$-BLOSSOMING

“…Some of these properties, algorithms and identities were also derived using the standard mathematical induction and other elementary techniques in [4] by Jegdić, Larson, and Simeonov. Recently, Jegdić, Simeonov, and Zafiris used the tensor product and generalized concept of q-blossoming for polynomials in one variable introduced in [15] to define qblossoming for polynomials in two variables leading to the study of q-Bézier surfaces in [5].…”