دانلود رایگان مقاله کاهش چند درجه محدود با توجه به نورم ژاکوبی

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).

5. Conclusion

We have derived an analogous result to Lutterkort et al. (1999) for solving the problem of multi-degree reduction of polynomials with boundary constraints with respect to Jacobi inner product. We have proved that the best constrained degree-reduced approximation is equal to the weighted least squares fit of the Bézier coefficients with factored Hahn weights. We have shown on several examples that the two additional parameters of the Jacobi L2-norm, when compared to the classical L2 inner product, can serve as supplementary shape parameters for constrained degree reduction.