Geometric Construction Method of G² Lupaş q-Beta Spline Curves

 Lupaş q-Bézier curves are a family of generalized Bézier curves with rational function as the basis function. To enhance the local shape control capabilities of composite Lupaş q-Bézier curves, a geometric construction method for G² continuous Lupaş q-Beta spline curves is proposed. According to the endpoint properties of Lupaş q-Bézier curve and the Beta constraint conditions, the necessary and sufficient conditions for the G² continuity of the combined curve are deduced. Then, G² Lupaş q-Beta spline curves at the splicing point are constructed by geometric algorithm. Moreover, we discuss the influence of shape parameters on the shape of Lupaş q-Beta spline curve from the perspective of algebra and geometry, respectively. G² Lupaş q-Beta spline curves surpass C¹G² Lupaş q-Gamma spline curves, which dependened on global parameter segmentation. Numerical examples results indicate that, compared to classical Beta spline curves, Lupaş q-Beta spline curves offer greater modeling advantages and flexibility.