Geometric Degree Reduction of Wang–Ball Curves

Abstract

There are substantial methods of degree reduction in the literature. Existing methods share some common limitations, such as lack of geometric continuity, complex computations, and one-degree reduction at a time. In this paper, an approximate geometric multidegree reduction algorithm of Wang–Ball curves is proposed. $G^0$ -, $G^1$ -, and $G^2$ -continuity conditions are applied in the degree reduction process to preserve the boundary control points. The general equation for high-order (G2 and above) multidegree reduction algorithms is nonlinear, and the solutions of these nonlinear systems are quite expensive. In this paper, $C^1$ -continuity conditions are imposed besides the $G^2$ -continuity conditions. While some existing methods only achieve the multidegree reduction by repeating the one-degree reduction method recursively, our proposed method achieves multidegree reduction at once. The distance between the original curve and the degree-reduced curve is measured with the $L_2$ -norm. Numerical example and figures are presented to state the adequacy of the algorithm. The proposed method not only outperforms the existing method of degree reduction of Wang–Ball curves but also guarantees geometric continuity conditions at the boundary points, which is very important in CAD and geometric modeling.