Lupaş blending functions with shifted knots and q-Bézier curves

Abstract

Abstract In this paper, we introduce blending functions of Lupaş q-Bernstein operators with shifted knots for constructing q-Bézier curves and surfaces. We study the nature of degree elevation and degree reduction for Lupaş q-Bézier Bernstein functions with shifted knots for t ∈ [ a [ μ ] q + b , [ μ ] q + a [ μ ] q + b ] $t \in [\frac{a}{[\mu ]_{q}+b} , \frac{[\mu ]_{q}+a}{[\mu ]_{q}+b} ]$ . For the parameters a = b = 0 $a=b=0$ , we get Lupaş q-Bézier curves defined on [ 0 , 1 ] $[0,1]$ . We show that Lupaş q-Bernstein functions with shifted knots are tangent to fore-and-aft of its polygon at end points. We present a de Casteljau algorithm to compute Bernstein Bézier curves and surfaces with shifted knots. The new curves have some properties similar to q-Bézier curves. Similarly, we discuss the properties of the tensor product for Lupaş q-Bézier surfaces with shifted knots over the rectangular domain.

Keywords

• q-integers
• Degree elevation
• De Casteljau-type algorithm
• Lupaş q-Bernstein operators with shifted knots
• Bézier curve
• Tensor product