Multi－degree reduction of NURBS curves based on their explicit matrix representation and polynomial approximation theory

• Published in: Science China. Information sciences Vol. 47; no. 1; pp. 44 - 54
• Main Author: CHENG, Min
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.02.2004; Institute of Images and Graphics, State Key Laboratory of CAD & CG, Zhejiang University, Hangzhou 310027, China
• Subjects: Approximation; CAD; Chebyshev approximation; Computer graphics; Degree reduction; Design engineering; Mathematical analysis; Matrix representation; NURBS曲线; Polynomials; 几何学; 圆锥曲线; 多项式近似; NURBS curves; matrix representation; Chebyshev polynomials; multi-degree reduction
• ISSN: 1009-2757; 1674-733X; 1862-2836; 1869-1919
• DOI: 10.1360/02yf0229

NURBS curve is one of the most commonly used tools in CAD systems and geometric modeling for its various specialties, which means that its shape is locally adjustable as well as its continuity order, and it can represent a conic curve precisely. But how to do degree reduction of NURBS curves in a fast and efficient way still remains a puzzling problem. By applying the theory of the best uniform approximation of Chebyshev polynomials and the explicit matrix representation of NURBS curves, this paper gives the necessary and sufficient condition for degree reducible NURBS curves in an explicit form.And a new way of doing degree reduction of NURBS curves is also presented, including the multi-degree reduction of a NURBS curve on each knot span and the multi-degree reduction of a whole NURBS curve. This method is easy to carry out, and only involves simple calculations. It provides a new way of doing degree reduction of NURBS curves,which can be widely used in computer graphics and industrial design.