Optimal multi-degree reduction of Bézier curves with constraints of endpoints continuity

Given a Bezier curve of degree n, the problem of optimal multi-degree reduction (degree reduction of more than one degree) by a Bezier curve of degree m (m > n - 1) with constraints of endpoints continuity is investigated. With respect to L2 norm, this paper presents one approximate method (MDR by L2) that gives an explicit solution to deal with it. The method has good properties of endpoints interpolation: continuity of any r, s (r, s ≥ 0) orders can be preserved at two endpoints respectively. The method in the paper performs multi-degree reduction at one time and does not need the stepwise computing. When applied to the multi-degree reduction with endpoints continuity of any orders, the MDR by L2 obtains the best least squares approximation. Comparison with another method of multi-degree reduction (MDR by L∞), which achieves the nearly best uniform approximation with respect to L∞ norm, is also given. The approximate effect of the MDR by L2 is better than that of the MDR by L∞. Explicit approximate error analysis of the multi-degree reduction methods is presented.