Degree reduction of Bézier curves by \(L^ 1\)-approximation with endpoint interpolation
In this paper the one-degree reduction problem with endpoint interpolation in the \(L^1\)-norm is considered. The best one-degree reduction of Bezier curve of the degree \(n \leq 5\) with endpoint interpolation by using the perfect splines is obtained. For the higher degree, it is practically impossible to get the best one-degree reduction in closed form. For the general degree \(n\), it is proposed a ``good'' one degree reduction by use of an appropriate transform of the Chebychev polynomials \(U_n(x)\) of the second type of degree \(n\). By use of the good one-degree reduction, subdivision algorithm is given to get one-degree reduced Bézier curve within a given tolerance \({\mathcal E}\). Let \(\Pi_n[0,1]\) be the set of polynomials \(f_n\) degree \(n\) such that \(f_n(0)= 0=f_n(1)\). A perfect spline of degree \(n\) with \(r\) knots \(0<u_1< \cdots < u_r <1\) on \([0,1]\) is a function of the form \[ P_n(x) =c\left(x^n+2 \sum^r_{i=1} (-1)^i(x-u_i)^n_+ \right) + \sum^{n-1}_{j=0} a_jx^j, \] where \(c\), \(a_0, \dots, a_{n-1}\) are real constants. We say that \(P_n^{n-2} [0,1]\) is the set of perfect splines of degree \(n\), satisfying \(P^i_n(0) =P^i_n(1)\), \(i=0,1, \dots, n-2\), \(|P^n(x) |=1\) and that it corresponds to \(\Pi_n [0,1]\). The following two results are characteristic for this paper: I. A polynomial \(p_{n-1}\) of degree \(n-1\) is the best \(L^1\)-approximation to a given polynomial \(f_n\) of degree \(n\) with endpoint interpolation, if and only if there exists a perfect spline \(P_n\in P_n^{n-2} [0,1]\), where \(P_n^{n -2} [0,1]\) corresponds to \(\Pi_n [0,1]\) such that \(P_n^{(n)} (x)= \text{sgn} (f_n- p_{n-1})\). II. For a Bézier curve \(f_3\) of degree 3 with \(\Delta^3b_0 \neq 0\), the best one degree reduction \(p_2\) of \(f_3\) is given as \(p_2(t)= \sum^2_{l=0} \overline b_iB^2_i(t)\) where \(\overline b_0= b_0\), \(\overline b_1= -{1\over 4} b_0+ {3\over 4} b_1+ {3\over 4} b_1-{1 \over 4} b_3\), \(\overline b_2 =b_3\). The \(L^1\)-norm error is given by \(|\Delta^3 b_0 |/32\).
scientific article; zbMATH DE number 1021988