Discrete convolution generation of the stirling curve and its evaluation algorithm(斯特林曲线的离散卷积生成及其求值算法)

Abstract

The Stirling basis function is a kind of rational basis function generated by discrete probability model. By analyzing the layer-by-layer recurrence relation of the basis functions, the nth degree Stirling basis functions sequence is generated via discrete convolution. For the Stirling curve, n! de Casteljau algorithms are obtained for the recursive evaluation of curves. Furthermore, we obtain two evaluation algorithms with linear complexity, as well as the hodograph of the curve with discrete convolution representation and the explicit expression for the derivation of the first and last basis functions. The method in this paper can be extended to the study of the rational basis functions, curves and surfaces on a class of nested spaces.(斯特林基函数是由离散概率模型生成的一类有理基函数。通过分析基函数的逐层递推关系，构造了斯特林基函数的离散卷积结构。结合离散卷积满足的交换性，得到n次斯特林曲线的n！种de Casteljau算法，并将其用于曲线的递归求值，进而得到n次斯特林曲线的2种线性求值算法、速端曲线离散卷积表示以及首末两个n次斯特林基函数的导函数显式表达式。研究可推广至一类嵌套空间中的有理基函数及其曲线曲面。)

Keywords

• stirling curves(斯特林曲线)
• discrete convolution(离散卷积)
• de casteljau algorithm(de casteljau算法)
• linear complexity(线性复杂度)
• hodograph(速端曲线)