Weighted Lupaş qq–Bézier curves

This paper is concerned with a new generalization of rational Bernstein–Bézier curves involving qq-integers as shape parameters. A one parameter family of rational Bernstein–Bézier curves, weighted Lupaş qq–Bézier curves, is constructed based on a set of Lupaş qq-analogue of Bernstein functions which is proved to be a normalized totally positive basis. The generalized rational Bézier curve is investigated from a geometric point of view. The investigation provides the geometric meaning of the weights and the representation for conic sections. We also obtain degree evaluation and de Casteljau algorithms by means of homogeneous coordinates. Numerical examples show that weighted Lupaş qq–Bézier curves have more modeling flexibility than classical rational Bernstein–Bézier curves and Lupaş qq–Bézier curves, and meanwhile they provide better approximations to the control polygon than rational Phillips qq–Bézier curves.