Approximate multi-degree reduction of Q-Bézie curves via generalized Bernstein polynomial functions

Abstract

Abstract Quasi Bézier curves (or QB-curves, for short) possess the excellent geometric features of classical Bézier curves and also have good shape adjustability. In this paper, an algorithm for a multi-degree reduction of QB-curves based on L 2 $L_{2}$ norm and by the analysis of geometric characteristics of QB-curves is constructed. The approximating approach for QB-curves of degree n + 1 $n+1$ by degree m ( m ≤ n $m\leq n$ ) is also given. Secondly, by solving the linear equations under the constraints of C 0 $C^{0}$ and C 1 $C^{1}$ and without constraints, the explicit expression of the points of the approximating curve is obtained, which minimizes the error between the original curve and the approximating curve using the least square method. Some numerical examples of degree reduction under different constraints are given, and the corresponding errors are calculated as well. The results show that this method can be easily implemented, is highly precise and very effective.

Keywords

• Generalized Q-Bézier basis functions
• Q-Bézier curve
• Degree reduction
• L 2 $L_{2}$ norm
• Least squares approximation