1. Introduction
Optimal degree reduction is one of the fundamental tasks in Computer Aided Geometric Design (CAGD) and therefore has attracted researchers’ attention for several decades (Ait-Haddou, in press; Zhou and Wang, 2009; Ahn et al., 2004; Ahn, 2003; Kim and Ahn, 2000; Lutterkort et al., 1999; Watkins and Worsey, 1988). Used not only for data compression, CAD/CAM software typically requires algorithms capable of converting a curve (surface) of a high degree to a curve (surface) of a lower degree. Considering the problem coordinate-wise, the goal is formulated as follows: given a univariate polynomial p of degree n, find its best polynomial approximation q of degree m, m < n, with respect to a certain given norm. The degree reduction can be seen as an inverse operation to the degree elevation. Whereas elevating polynomial degree from m to n is always possible, see e.g. (Hoschek and Lasser, 1993), because it is equivalent to expressing a polynomial q ∈ Pm in the basis of a larger linear space Pn, Pm ⊂ Pn, the degree reduction is in general not. A natural alternative is then finding the best approximation that minimizes a certain error. This can be interpreted as projecting p ∈ Pn into Pm. Depending on a particular norm defined on Pn, various schemes for degree reduction were derived (Eck, 1993; Peters and Reif, 2000; Lee and Park, 1997; Kim and Moon, 1997; Brunnett et al., 1996; Ait-Haddou and Goldman, 2015).
