Optimal Multi-degree Reduction Of A New Curve That Has Shape Preserving And Linear Evaluation Properties

In 2003 Delgado and Pena introduced a new generalized Ball curve. Due to the fast evaluation with linear complexity and the shape preserving property for this generalized Ball curve, this type of curve provides a new direction for geometry design. Its main idea is to construct a new normalized totally positive basis, and make sure that the degrees are distributed like a canyon according to the respective sequence number, high in both sides and low in middle. Thus it overcomes the shortage of Wang-Ball curve which has linear complexity but not the NTP basis and Said-Ball curve which belongs to the NTP basis but does not have the linear complexity. For convenience, this type of curve is called DP-NTP curve, and its basis called univariate DP-NTP basis.Based on the advantages of the DP-NTP basis, it is significant and valuable to develop new effective algorithms, including the degree reduction algorithms which play an important role in the data communications and data compression. According to the elementary property and degree attenuation of univariate DP-NTP basis, this paper gives two constrained multi-degree reduction approximation algorithms in the explicit expression as follows:One is to apply the generalized inverse of the evolution matrix of the curve to accomplish the endpoints constrained multi-degree reduction.The other is to translate the DP-NTP basis to the Jacobi basis to accomplish the constrained multi-degree reduction of the curve.We give numerical experiments and the error analysis, and thus provide a reliable reference for the Engineering staffs. The result has great contributions and effect on the improvements of the computer aided design system.