Optimal Constrained Explicit Multi-degree Reduction Approximation of DP Curves ★

DP curves, which possess the shape-preserving property and linear complexity evaluation algorithm, occupy a significantly important place in computer aided geometric design. Thus to do the research on the DP curves algorithms has a positive meaning. Aphirukmatakun et al. presented a degree elevation algorithm of DP curves based on the transformation between the DP basis and power basis. At the same time, Itsariyawanich et al. put forward a degree reduction algorithm. However, up to now, there has been no optimal algorithm for explicit multi-degree reduction of DP curves with endpoints constraints. In this paper, based on the orthogonality of Jacobi basis, we give out an effective algorithm of constrained multi-degree reduction approximation of DP curve. We prove that the approximation is optimal in the L2 norm. In the end of this paper, we demonstrate the validity and efficiency of our algorithm.